Positive Operators and Semigroups on Banach Lattices: Proceedings of a Caribbean Mathematics Foundation Conference 1990

POSITIVE OPERATORS AND SEMIGROUPS ON BANACH LATTICES Proceedings of a Caribbean

Mathematics Foundation

Conference 1990

Edited by

C. B. HUIJSMANS Department of Mathematics and Computer Science, Leiden University, The Netherlands

and

W. A. J. LUXEMBURG California Institute a/Technology, Pasadena. U.S.A.

Reprinted from Acta Applicandae Mathematicae, Vol. 27, Nos. 1-2 (1992)

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

?osltlve operators and semlgroups on Banach lattlces the Carlbbean Mathematlcs Foundatl0n's conference. C.S. HU1Jsmans and W.A.J. Luxemburg.

p. cm.

proceedlngs of 1990 / edlted by

1. Posltlve operators--Congresses. 2. Semlgroups of opera tors-Corgresses. 3. Banach lattlces--Congresses. I. HU1Jsmans. C. B. II. Luxemburg. W. A. J .• 1929- III. Carlbbean Mathematlcs Foundatl0n. QA329.2.P67 1992 5'5' .7242--dc20

ISIJN 978-90-481-4205-7

Printed on acid-free paper

Ali Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992

92-26747

No part of the material protected by this copyright notice may be reproduced Of

utilized in any fonn Of by any means, electronic Of mechanical, includ ing photocopying, recording Of by any infonnation storage and retrieval system, without written pennission from the copyright owner.

ISBN 978-90-481-4205-7 ISBN 978-94-017-2721-1 (eBook)DOI 10.1007/978-94-017-2721-1

Table of Contents

Preface

List of Participants

Y. A. ABRAMOVICH, C. D. ALIPRANTIS, and O. BURKINSHA W / Positive Operators on Krein Spaces

Y. A. ABRAMOVICH and W. FILTER / A Remark on the Representation of

v

VB

Vector Lattices as Spaces of Continuous Real-Valued Functions 23

W. ARENDT and J. VOIGT / Domination of Uniformly Continuous Semigroups 27

S. J. BERNAU / Sums and Extensions of Vector Lattice Homomorphisms 33

B. EBERHARDT and G. GREINER / Baillon's Theorem on Maximal Regularity 47

A. W. HAGER and J. MARTINEZ / Fraction-Dense Algebras and Spaces 55

C. B. HUIJSMANS and W. A. J. LUXEMBURG / An Alternative Proof of a Radon-Nikodym Theorem for Lattice Homomorphisms 67

C. B. HUIJSMANS and B. DE PAGTER / Some Remarks on Disjointness Preserving Operators 73

L. MALIGRANDA / Weakly Compact Operators and Interpolation 79

P. MEYER-NIEBERG / Aspects of Local Spectral Theory for Positive Operators 91

B. DE PAGTER / A Wiener-Young Type Theorem for Dual Semigroups 101

A. R. SCHEP / Krivine' s Theorem and Indices of a Banach Lattice 111

A. W. WICKSTEAD I Representations of Archimedean Riesz Spaces by Con-tinuous Functions 123

X.-D. ZHANG / Some Aspects of the Spectral Theory of Positive Operators 135

Problem Section 143

Acta Applicandae Mathematicae 27: v-vi, 1992. © 1992 Kluwer Academic Publishers.

Preface

v

During the last thirty years advances in the theory of ordered algebraic structures such as vector lattices (Riesz spaces), f -algebras and Banach lattices have played a very important role in the recent development of the theory of positive linear operators that has its roots in the fundamental results of Frobenius and Perron about the spectral properties of positive matrices. Moreover, motivated by problems concerning partial differential equations, particularly those dealing with initial value problems, probability theory (Markov processes), mathematical physics and control theory, the theory of one-parameter semi groups of positive linear operators on 'Banach lattices has undergone a tremendous growth during the last decades.

From June 18 through June 22, 1990 on the Caribbean island of Cura<;ao (Netherlands Antilles) a small workshop was held devoted to the theory of positive operators and their semigroups. Following the workshop a conference was held from June 25-June 29 primarily on recent advances in this area of positive operators. The workshop and the conference took place under the auspices of the Caribbean Mathematical Foundation (CMF) under the directorship of Dr J. Martinez.

The purpose of the workshop, conducted by C.B. Huijsmans, W.AJ. Luxemburg and B. de Pagter, was to present to a group of interested mathematicians from the Caribbean and Latin America an up-to-date account of the main results of the theory of positive operators on Banach lattices. The workshop was attended by mathematicians from Florida U.S.A., Guyana, Panama, Surinam and Venezuela. There were three one and a half hours sessions per day during five days.

The conference following the workshop was organised by C.B. Huijsmans and W.A.J. Luxemburg with the main purpose to bring together a group of likeminded specialists from the U.S.A. and Western Europe to present their recent results and to discuss their research interests. The worked-out versions of the papers that were presented at the conference and some related contributions are collected in these proceedings. All the submitted articles were refereed.

We take this opportunity to thank all the participants of both the workshop and the conference for their contributions which made this mathematical gathering so successful.

The financial contribution by CMF (supported by NSF, the University of

VI PREFACE

Florida, de Universiteit van de Nederlandse Antillen, Fondo Fundashon Universidat and het Bestuurscollege Eilandsgebied Curas,:ao) is greatly appreciated by all the participants. Notably, we wish to express our sincere gratitude to the director of the CMF, Dr J. Martinez, for his encouragement, continuous support and participation. Without his unceasing efforts this workshop and conference would never have taken place. Finally, we thank the editorial staff of Kluwer Academic Publishers for their cooperation and support during the preparation of this manuscript.

Leiden/pasadena, Summer 1992 C.B. HUIJSMANS, W.A.J. LUXEMBURG

Acta Applicandae Mathematicae 27: 1992. Vll

List of Participants

C.D. ALIPRANTIS, Department of Mathematics, IUPUI, 1125 East 38th Street, Indianapolis, Indiana 46205-2810, U.S.A.

W. ARENDT, Equipe de MatMmatiques, Universite de Franche-Comte, 25030 Besan<;:on Cedex, France.

SJ. BERNAU, Department of Mathematical Sciences, The University of Texas at EI Paso, EI Paso, Texas, 79968-0514, U.S.A.

G. GREINER, Mathematisches Institut, UniversiHit Tiibingen, Auf der Morgenstelle 10, D-4700 Tiibingen 1, Germany.

C.B. HUIJSMANS, Mathematisch Instituut, Rijksuniversiteit Leiden, P.O. Box 9512, Niels Bohrweg 1,2300 RA Leiden, The Netherlands.

W.A.J. LUXEMBURG, Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125, U.S.A.

L. MALIGRANDA, Departamento de Matematicas, IVIC, Apartado 21827, Cara<;:as 1020-A, Venezuela.

J. MARTINEZ, Department of Mathematics, University of Florida, 201 Walker Hall, Gainesville, Florida 32611-2082, U.S.A.

P. MEYER-N lEBERG, Fachbereich Mathematik, Universitat Osnabriick, AlbrechtstraSe 28,4500 Osnabrock, Postfach 4469, Germany.

B. de PAGTER, Faculteit der Wiskunde en Informatica, Mekelweg 4, Postbus 5031,2600 GA Delft, The Netherlands.

A.R. SCHEP, Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A.

A.W. WICKSTEAD, Department of Pure Mathematics, The Queen's University of Belfast, Belfast BTI INN, Northern Ireland, u.K.

X-D. ZHANG, Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125, U.S.A.

Acta Applicandae Mathematicae 27: 1-22, 1992. © 1992 Kluwer Academic Publishers.

Positi ve Operators on Krein Spaces

Dedicated to the memory of M.G. Krein (1907-1989)

Y.A. ABRAMOVICH, C.D. ALIPRANTIS and O. BURKINS HAW Department of Mathematics, IUPUl, Indianapolis, IN 46205, U.S.A

(Received: 27 April 1992)

Abstract. A Krein operator is a positive operator, acting on a partially ordered Banach space, that carries positive elements to strong units. The purpose of this paper is to present a survey of the remarkable spectral properties (most of which were established by M.G. Krein) of these operators. The proofs presented here seem to be simpler than the ones existing in the literatnre. Some new results are also obtained. For instance, it is shown that every positive operator on a Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace.

Mathematics Subject Classifications (1991): 46C50, 47B65, 47B37

Key words: partially ordered Banach space, Krein space, Krein operator, hyperinvariant subspace

1. Introduction

The classical theorem of M.G. Krein and M.A. Rutman asserting that the spectral radius of a nonquasinilpotent positive compact operator is an eigenvalue having a positive eigenvector is well known and widely used in the literature. This result was just one of the many important results concerning spectral properties of positive operators that appeared in [8]. It seems that most of these results are not well known and their proofs are not readily available. The purpose of this paper is to present a survey (with proofs) of these results in the setting of partially ordered Banach spaces. This gives us the opportunity to present at the same time some of the basic order properties of partially ordered normed spaces. Due to our advantage of hindsight, many of our proofs are simplified versions of existing proofs in the literature and are presented in a systematic way using modern techniques and terminology.

We are especially interested in the intriguing class of operators that carry positive elements to order units. We shall call these operators Krein operators. Some remarkable spectral properties of Krein operators are discussed here and a variety of examples illustrate their usefulness and importance.

In general, the paper presents results that deal with eigenvalues and fixed

2 Y.A. ABRAMOVICH, C.D. ALIPRANTIS AND O. BURKINS HAW

points of positive operators (or families of positive operators) or their adjoints. Most of these beautiful results are due to either M.G. Krein and his collaborators or they have been influenced by his work. Accordingly, we dedicate the paper to the memory of this great mathematician.

2. Partially Ordered Normed Spaces

Recall that a real vector space X equipped with a partial order 2: is said to be a partially ordered vector space or simply an ordered vector space whenever

1) x 2: y implies ax 2: ay for all a 2: 0; and 2) x ~ y implies x + z ~ y + z for all Z E X.

The positive cone (or simply the cone) X+ of X is the set of all positive elements of X, i.e., X+ = {x E X: x ~ O}. The cone X+ is said to be generating whenever X = X+ - X+, i.e., whenever every vector can be written as a difference of two positive vectors.

A partially ordered vector space is said to be Archimedean whenever nx ::; y for each n and some x, y imply x ::; O. Notice that in an Archimedean partially ordered vector space y 2: 0 and -EY ::; X ::; Ey for all E > 0 imply x = O.

LEMMA 2.1 If a partially ordered vector space X admits a Hausdoif.! linear topology for which X+ is closed, then X is Archimedean.

Proof. Let l' be a Hausdorff linear topology on a partially ordered vector space X such that X+ is T-closed. Now assume nx ::; y for all n and some x, y E X. Then *y - x 2: 0 and *y - x ~ -x imply -x 2: 0 or x ::; o. Q.E.D.

THEOREM 2.2 (M.G. Krein-Smulian) If X is a Banach space ordered by a closed generating cone, then there exists a constant J'v1 > 0 such that for each x E X there are vectors Xl, X2 E X+ satisfying

x = Xl - X2 and Ilxill ::; Mllxll (i = 1,2).

Proof. The proof below is due to B.Z. Vulikh [15] and closely resembles the proof of the open mapping theorem. For each real number t 2: 0 we define the set

Et = {x EX: :3Xl,X2 E X+ with x = Xl - X2 and Ilxill ::; t (i = 1,2)}.

Clearly, each Et is convex, symmetric, and 0 E E t. In addition, note that aEt = Eat holds for each a 2: 0, and 0 ::; s ::; t implies Es S;;; E t.

Since X+ is a generating cone, we see that X = U~=l En. So, by the Baire Category Theorem, there exist a natural number k, an Xo E X and an r > 0 such that the closed ball C(xo, r) satisfies

C(xo, r) = {x EX: Ilxo - xii::; r} S;;; Ek.

POSITIVE OPERATORS ON KREIN SPACES 3

From the symmetry of Ek we see that Ilxll ::::; r implies Xo + x, x - Xo E Ek. and so by the convexity of Ek we infer that x = ~(xo + x) + ~(x - xo) E Ek.

Therefore, C(O, r) ~ Ek .

We claim that C(O, r) ~ E2k holds. If this is established, then clearly for x E X there exist XI,X2 E X+ with x = XI-X2 and Ilxill ::::; 2:llxll for i = 1,2.

To establish that C(O, r) ~ E2k holds, let x E C(O, r). Then ~x E Ek. So

there exists an Xl E Ek such that II~x - XIII < i. Note that IlxI11 ::::; rand

Ilx - xIII = II (~X - Xl) + ~xll < ~ + ~ = &. So, X-Xl E C (0, 1) = !C(O, r) ~ Ef£. Repeating the same argument and using

2

X - Xl instead of x, we see that there exists an X2 E E!:£ such that IIx211 ::::; 1 2

and Ilx - Xl - x211 < i. So, by an inductive argument, we see that there exists a sequence {xn} in X such that 1) Xn E E_k_ and Ilxnll ::::; 2LI for n = 1,2, ... ; and

2n - 1

2) Ilx - I:~l xiii < {n. Now for each n choose Yn, Zn E X+ such that Xn = Yn - Zn, IIYnl1 ::::; 2nk_I'

and Ilznll ::::; 2nk_ l . If Y = I:~=l Yn and Z = I:~=l Zn, then (since X+ is norm closed) y, Z E X+. From I:~l IIYnl1 ::::; 2k and I:~=l Ilznll ::::; 2k we see that Y - Z E E2k. Finally, from 2) we get X = I:~l Xi = Y - Z E E2k. and the proof is finished. Q.E.D.

COROLLARY 2.3 Let X be a Banach space partially ordered by a closed generating cone. If Xn -----+ X holds in X, then there exist two sequences {Yn} and {zn} in X+ satisfying 1) Yn -----+ Y and Zn -----+ z; 2) Xn = Yn - Zn for each n; and 3) X = Y - z.

Proof Using Theorem 2.2, we know that there exist an M > ° and two sequences {an} and {bn} in X+ satisfying Xn -x = an -bn, Ilanll ::::; Mllxn -xii, and Ilbnll ::::; Mllxn -xii for each n. So, an -----+ ° and bn -----+ 0. Now write X = y-z with y, Z E X+, then let Yn = y+an and Zn = z+bn and note that the sequences {Yn} and {zn} satisfy the desired properties. Q.E.D.

COROLLARY 2.4 Let X be a Banach space partially ordered by a closed generating cone and let Y be a topological vector space. Then an operator T: X -----+ Y is continuous if and only if T: X+ -----+ Y is continuous.

An operator T: X -----+ Y between two ordered vector spaces (where, as usual, 'operator' means 'linear operator') is said to be positive whenever X ?: ° implies T(x) ?: 0.

4 Y.A. ABRAMOVICH, C.O. ALIPRANTIS AND O. BURKINSHAW

It is remarkable that quite often positive operators are automatically continuous. This was first proved by M.G. Krein for positive linear functionals [8] and later was generalized in several contexts by various authors; see, for instance, [3], [10],[11], and [14]. The next result, due to G.Ya. Lozanovsky, is the strongest in this direction and appeared in [16].

COROLLARY 2.5 (Lozanovsky) Let X and Y be two partially ordered Banach spaces with closed cones. If the cone of X is also generating, then every positive operator T: X -+ Y is continuous.

Proof It suffices to show that the operator T has a closed graph. So, assume Xn -+ 0 in X and TXn -+ Y in Y. By passing to a subsequence, we can also assume that L~=l nllxnll < 00. By Theorem 2.2 there exist an M > 0 and two sequences {Yn} and {zn} in X+ satisfying Xn = Yn - Zn, IIYnl1 :s: Mllxnll, and Ilznll :s: Mllxnll for each n. Since X+ is closed, the vector Z = L~=l n(Yn +zn) in X belongs to X+, and -z :s: nXn :s: Z holds for each n. From the positivity of T we infer that -~Tz :s: TXn :s: ~Tz. Using that Y+ is also closed, we conclude that 0 S y :s: O. That is, y = 0 and the proof is finished. Q.E.D.

3. Krein Spaces

Let X be a partially ordered vector space. A vector u E X+ is said to be a strong unit (or simply a unit) whenever for each x E X there exists an a > 0 such that x S au. The set of all units in X will be denoted by U. Clearly,

and aU = U for all a > O.

If an ordered vector space has a unit, then it is clear that its cone is generating.

DEFINITION 3.1 A partially ordered Banach space X is said to be a Krein space whenever

a) X+ is closed; and b) X has strong units, i.e., U ic 0.

Notice that every Krein space is automatically Archimedean and its cone is generating. Since the cone of a Krein space is closed, its order intervals are likewise closed sets. Here are some examples of Krein spaces.

The classical Banach lattices C(K) of all real-valued continuous functions on a Hausdorff compact topological space K. The algebraic, order, and lattice operations are defined pointwise and the norm is the sup norm. The constant function 1 is a unit. Incidentally, the reader can convince himself that the spaces C(K) are the only Banach lattices that are Krein spaces. The vector space Ck[a, b] of all k-times continuously differentiable realvalued functions on a (bounded) closed interval [a, b]. The algebraic and order operations are defined pointwise. The norm is defined by

Ilxll = Ilxlloo + Ilx'lloo + ... + Ilx(k) 1100'


where II . 1100 denotes the sup norm. The constant function 1 is a unit. The vector space 8(H) of all bounded operators on a complex Hilbert space H. Here we consider 8(H) as a real vector space. The ordering of 8(H) is defined by saying T ~ S whenever (Tx, x) ~ (Sx, x) for all x E H. The identity operator I is a unit.

In a Krein space the strong units are characterized as follows.

LEMMA 3.2 For an element u > ° in a Krein space X the following assertions are equivalent. 1) The element u is a strong unit. 2) For each x E X there exists some ao such that u + ax E X+ holds for all

lal ::; ao· 3) The vector u is an interior point of X+.

In particular, in every Krein space X the set of all units U = Int( X +) is norm dense in X+

Proof 1) ===? 2) Let x E X. Pick some (3 > ° such that ±x ::; (3u. Put ao = ~ and note that u + ax ~ ° holds for all lal ::; ao.

2) ===? 3) From our hypothesis it follows that

00 00 X = U [-nu,nuj = U n[-u,uj.

n=l n=l

Since X is a Banach space, it follows from the Baire Category Theorem that the order interval [-u, u j has an interior point, say a. So there exists an open ball B(a,r) such that B(a,r) ~ [-u,u]. We claim that B(u,r) ~ X+.

To see this, let x E B(u, r), i.e., let Ilx-ull < r. Then a+x-u and a-x +u belong to B(a, r) and so -u ::; a + x - u and a - x + u ::; u both hold. So a + x ~ ° and x - a ~ 0, and by adding we see that 2x ~ 0, or x ~ 0, i.e., x E X+. Therefore, B(u, r) ~ X+.

3) ===? 1) Assume that u is an interior point of X+. So there exists a neighborhood V of zero such that u + V ~ X+. Now let x E X. Pick some a > ° such that -ax = a( -x) E V. Then u - ax ~ ° and so x ::; ±u, which shows that u is a unit.

For the last part note that if x E X+ and u E U, then x + *u E U holds for each n, and x + *u --+ x. Q.E.D.

One more property of Krein spaces is stated in the following lemma.

LEMMA 3.3 Let X be a Krein space and let x E X+. Then for each y Ie ° there exists a scalar a such that x + ay E 8X+.

Proof If x E 8X+, then a = ° is the desired constant. So we can assume that x E Int(X+). Let

(3 = inf{ a E R: x + ay E X+} and 1= sup { a E R: x + ay E X+}.


Since x E Int(X+), we see that f3 < 0 < I' We claim that either f3 or 1 is finite. Indeed, if f3 = -00 and 1 = 00, then (by the convexity of X+) it follows that x + ay E X + for all a E R. In particular, we have ±ny ::; x for each n. Since X is Archimedean, we infer that ±y ::; 0 or y = 0, which is a contradiction.

Now note that if f3 E R, then x + f3y E ax+, and similarly if 1 E R, then x + IY E ax+. Q.E.D.

Now consider a Krein space X and let u be a strong unit in X. Then the formula

Ilxll u = inf{t > 0: -tu::; x ::S tu}

defines a norm on X. (The fact that Ilxllu = 0 holds if and only if x = 0 follows from the Archimedean property of X.) The norm I!·!!u will be referred to as the u-norm. The basic properties of the u-norm are included in the following result.

THEOREM 3.4 If u is a strong unit in a Krein space X, then: 1) The u-norm is monotone, i.e., O::S x ::S y implies Ilxll u ::S IlylJu. 2) There is an r > 0 such that rC ~ [-u, u], where C = {x E X: Ilx!! ::S I}. 3) There exists an M > 0 such that

Ilxl!u ::S M!lx!1 holds for all x E X.

4) Every positive linear functional on X is II . llu-continuous. 5) The cone X+ is also II . I/u-closed.

Proof 1) Straightforward. 2) The validity of this assertion follows from Lemma 3.2, 3). 3) Fix an r > 0 such that rC ~ [-u, u]. If x f= 0, then II~II E C, and .so

-u ::S II~II x ::S u, or _II~II u ::S x ::S II~II u. Therefore, I/xllu ::S II~II holds for each x EX.

4) Let f: X -+ R be a positive linear functional. From -I/xl/uu ::S x ::S I/xll u11

we see that -lIxllu f(u) ::; f(x) ::S IIxll uf(u), or If(x)1 ::S f(u)l/xl/ u, whicl shows that f is 1/ . lIu-continuous.

5) Assume that y tj:. X+. Since X+ is norm closed, by the classical separation theorem there exist f E X, and a constant c such that f (y) < c ::; f (x) for each x E X+. Since X+ is a cone, it is easy to see that c ::S 0 and that f(x) 2: 0 for each x E X+. So, f is a positive linear functional. By part 4), f is 1/ . 11ucontinuous. This shows that y cannot be in the " 'lIu-closure of X+, and so X+ is a II . I/u-closed set. Q.E.D.

4. The Dual of an Ordered Normed Space

Let X be a partially ordered normed space. As usual, the (norm) dual of X will be denoted by XI. On the dual Xl there is a natural order 2:, defined by saying

f 2: 9 whenever f(x) 2: g(x) holds for all x E X+.


If X+ is a generating cone, then it should be clear that X, equipped with?: is indeed a partially ordered Banach space. The positive cone of X, will be denoted by X~, i.e.,

X~ = {f E X': f(x)?: 0 for all x E X+},

and its members will be referred to as positive functionals; as usual, f > 0 means f ?: 0 and f i= O. The cone X~ will be referred to as the dual cone of X+.

LEMMA 4.1 If X is a partially ordered normed space, then the cone X~ is w* -closed (and hence norm closed).

Proof Let {fa} be a net in X~ such that fa ~ f, i.e., faex) -+ f(x) holds for each x E X. If x E X+, then fa(x) ?: 0 holds for each a, and so f(x) = lima fa (x) ?: 0. This shows that f E X~, and hence X~ is w*-closed. Q.E.D.

LEMMA 4.2 Let X be a partially ordered normed space whose positive cone X+ is norm closed. Then an element x E X is positive (i.e., x E X+) if and only if f(x) ?: ° holds for each f E x~.

Proof Assume that f(x) ?: ° holds for each f E X~. Also, suppose by way of contradiction that x ¢. X+. Then (by the separation theorem) there exist f E X, and a constant c such that f(x) < c :s: fey) holds for each y E X+.

Note that if y E X+, then fey) ?: ° must hold. (Otherwise, fey) < ° implies f(x) :s: f(ny) = nf(y) for each n, which is impossible.) Thus, f E X~ and so by our hypothesis f(x) ?: 0, which contradicts f(x) < c :s: f(O) = 0. Hence, x E X+ as desired. Q.E.D.

The dual cone of a Krein space need not be generating. The following example clarifies the situation.

EXAMPLE 4.3 Let X = el[O, 1]. As noticed before, the vector space el[O, 1] of all continuously differentiable functions is a Krein space under the norm

Ilxll = Ilxll oo + Ilx'll oo .

It is not difficult to see that this norm is equivalent to the norm

Illxlll = Ix(O)1 + Ilx'lloo.

It follows that the mapping J: X -+ REB e[O, 1], defined by J(x) = (x(O), x'), is a linear homeomorphism and so X can be identified with the Banach space Y = REBe[O, 1]. Transferring the cone X+ to Y, we see that Y becomes a Krein space under the cone

Y+ = {(x(O),x'): O:S: x E el[O, I]}.


Next, notice that Y' = REBM[O, 1], where M[O, 1] denotes the Banach space of all signed measures on [0, 1] of bounded variation. The pairing (Y, Y'j is given by

((a, z), ({3, p,) = a{3 + 101 z(t) dp,(t).

The dual cone Y~ of the cone Y+ is

Y~ = {({3, fl) : (3x(O) + fo1X'(t) dfl(t) 2: 0 for each O:S x E e1[0, I]}.

Notice that y' = ({3, p,) E Y~ implies (3 2: 0 and p, 2: O. Indeed, by letting first y = (1,0) E Y+, we get (3 2: o. To see that p, 2: 0, let 0 ~ z E e[o, 1] and consider x(t) = J~z(s)ds. Then 0 ~ x E el[O, 1] and hence y = (O,z) = (x(O),x') E Y+. Consequently, 0 ~ (y,y') = J~z(t)dp,(t), so that p, 2: O.

Now we claim that Y~ is not generating. To see this, denote by 80 the Dirac measure with support at 0 and consider (0,80) E Y'. Suppose by way of contradiction that there exist (a, p,), ({3, v) E Y~ such that (0,80 ) = ({3, v) - (a, p,). So, {3 = a and (a, 80 + p,) = (a, v) 2: 0 in Y'. If for each n we let xn(t) = (1- t)n, then 0 ~ Xn E e1[0, 1], and x~(t) = -n(1 - t)n-1 ~ o. So,

a - n = aXn(O) + x~(O)

2: aXn(O) + x~(O) + fo1 x~(t) dp,(t)

= ((xn(O), x~), (a, 80 + p,) 2: O.

Hence, n ~ a holds for each n, which is impossible. Consequently, the cone Y~ is not generating.

In Example 4.6 we shall prove a stronger result. Namely, that Y~ - Y~ (the vector subspace generated by Y~) is not even dense in Y'.

lt is worth investigating here when the dual cone X~ is generating. Recall that the cone X+ of a partially ordered normed space X is said to be normal whenever there exists a constant M > 0 such that 0 ~ x ~ y implies [[xli :S M[[y[[. Each of the following statements is equivalent to the normality of the cone.

o ~ Xn ~ Yn in X and Yn -+ 0 imply Xn -+ O. X admits an equivalent monotone norm, Le., X admits an equivalent norm III . III such that 0 :s x ~ y implies Illxlll :s Illylll·

For details see [5], [12, Chapt. 2], [15, Chapt. 4], and [17, Chapt. 5]. The important theorem relevant to our discussion is the following one due to M.O. Krein [7].

THEOREM 4.4 (M.O. Krein) In a partially ordered normed space X the cone X+ is normal if and only if the dual cone X~ is generating.


Proof Assume first that X+ is a normal cone and let f E X' satisfy Ilfll = 1. Pick some M > 1 such that ° ::; x ::; y implies Ilxll ::; MIIYII. Now consider the partially ordered Banach space Y = X EEl X EEl R and its vector subspace

Z = {(x, -x, - f(x)) : x EX}.

If B denotes the closed unit ball of X (i.e., if B = {x E X: Ilxll s I}), then the convex set A = B EEl B EEl [-M, M] satisfies

[Z + A - (O,O,SM)] n (X+ EElX+ EElR+) = 0.

Indeed, if there exist x EX, y, Z E Band -M < a < M such that x + y 2': 0, -x + Z 2': ° and - I (x) + a - S M 2': 0, then

Ilxll = II(x + y) - YII ::; Ilx + YII + Ilyll ::; Mlly + zll + Ilyll ::; 3M,

and so -f(x) = fe-x) ::; Ilfll·ll- xii::; 3M. Hence,

0::; - f(x) + a - SM ::; 3M + M - SM = -M < 0,

which is impossible. Since A has interior points, it follows from the classical separation theorem

that there exists a nonzero ¢ E y' such that ¢ 2': ° on X + ex + EEl R+ and ¢ ::; ° on Z + A - (0,0, SM). Since Z is a vector subspace, we infer that ¢ = ° on Z, or

¢(x, 0, 0) - ¢(O, x, 0) - ¢(O, 0, l)f(x) = ° for each x E X. Notice that). = ¢(O, 0,1) > ° and that g(x) = ¢(x, 0, 0) and hex) = ¢(O, x, 0) define positive continuous linear functionals on X. Consequently, I = tg - th holds. That is, I E X~ - X~, which shows that X~ is a generating cone.

For the converse, assume that X~ is generating. Then, by Theorem 2.2, there exists a constant M > ° such that every f E X' can be written in the form f = h - h with fi E X~ and Ilfi II ::; Mllfll for each i = 1,2. Now, let ° ::; x S yin X. If f E X' satisfies Ilfll ::; 1, then choose iI, 12 E X~ with IIIil1 s M (i = 1,2) and 1= h - 12, and note that

If(x)1 ::; iI(x) + hex) s hey) + h(y)s 2MIIYII·

Hence, Ilxll = sUPllfIl9If(x)1 ::; 2Mllyll, so that X+ is a normal cone. Q.E.D.

The reader should observe that since the cone Y+ of Example 4.3 is obviously not normal, Theorem 4.4 shows indirectly that the cone Y; is not generating.

The (norm) bidual X" of a partially ordered normed space X is the norm dual of XI, i.e., X" = (X')'. In the bidual X" we consider the binary relation 2': defined by

x" 2': y" whenever x" (x') 2': y" (x') holds for all x' E X~.

10 Y.A. ABRAMOVICH, C.O. ALIPRANTIS AND O. BURKINS HAW

In general, X" under the binary relation;::::: need not be a partially ordered Banach space. For this to happen it is necessary and sufficient that the vector subspace generated by X~ is norm dense in X'. We state this result as a lemma and leave its straightforward proof to the reader.

LEMMA 4.5 The bidual X" of a partially ordered normed space X is partially ordered (under ;:::::) if and only if X~ - X~ is norm dense in X'.

When X is a Krein space of the form C(D) for D Hausdorff and compact, we know that the bidual X" is again another Krein space of the type C(K). So, it seems natural to ask whether or not the bidual of a general Krein space is also a Krein space. Surprisingly, as the next example shows, the bidual of a Krein space need not be a Krein space.

EXAMPLE 4.6 Consider the Krein space X = C l [0, 1]. In Example 4.3, we saw that X can be identified with Y = R E9 C[O, 1] and the cone X+ with y+ = {(x(O), x'): ° :S x E C I [0, I]}. It was established there that y~ - y~ i= y/. In fact, we claim that a much stronger conclusion holds. Namely, y~ - y~ is not norm dense in y/. In order to prove this claim, note first that

Now let y/ = (r, J-L) E y~. As proved in Example 4.3 the measure J-L is a positive measure. We assert that J-L( {s}) = ° holds for each 0 :S S :S 1. We consider the case ° < s < 1. For each n, let Xn E C[O, 1] be as shown in Figure Ib). Obviously, an = Jdxn(t) dt < ° for each n and an --+ 0. Now let Yn = (-an, xn) and note that in view of (*) the sequence {Yn} belongs to y+. Therefore, (Yn, y/) ;::::: ° for each n. On the other hand, by the Lebesgue Dominated Convergence Theorem, we have

and so /-L( { S }) = 0. For the cases s = ° and s = 1 use the sequences shown in Figures Ia) and Ic).

To finish the proof, let I5s denote the Dirac measure supported at S E [0,1]. Then for each (aI, /-Lr) - (a2, /-L2) E y~ - y~, we have

II (0, I5s ) - [( aI, J-Lr) - (a2' J-L2)] II = max {ial - a21, Ill5s - /-LI + /-L211} ;::::: Il5s ({s}) -/-LI({S}) +/-L2({s})1 = 1,

which shows that (O,l5s ) tj. y~ - y~.


x x 0<5<1

5=1 5=0 L 1-

L n n n

"X= Xn(t) X = xn(t)

-1 ·1 ·1

Ca) (b) (e)

Fig. 1.

5. Krein Operators

We are now ready to introduce the notion of a Krein operator.

DEFINITION 5.1 A positive operator T: X -+ X on a Krein space is said to be a Krein operator whenever for each x > ° there exists some positive integer n (depending upon x) such that Tnx is a strong unit.

Before discussing the remarkable properties of Krein operators, let us present a few examples.

Our first example shows that in general Definition 5.1 does not imply that one can find a positive integer k for which Tkx is a strong unit for any vector x > 0. Consider the positive operator T: too -+ too defined by

Notice that if x = (0,0, ... ,0,Xn,Xn+l 1 "') E tt with Xn > 0, then Tnx is a strong unit while yn-1x is not a strong unit. Consequently, T is a Krein operator and there is no positive integer k such that Tkx is a strong unit for each x > 0. Let A = (aij) be a real n x n matrix such that aij > 0 holds for all (i, j). Then A, as an operator from Rn to Rn , is a Krein operator. The reader should notice that there are positive matrices that define Krein operators without having all their entries positive. For instance, the matrix

( ~ ~) defines a Krein operator on R2.

It is not difficult to see that a nonnegative n x n matrix defines a Krein operator if and only if for some k :2: 1 the matrix Ak has strictly positive entries. Let X be a Krein space, let u be a strong unit, and let f E X, be a strictly positive linear functional on X (Le., x > ° implies f(x) > 0). Then the rank-one operator T = f @ u is a Krein operator.


More generally, any finite rank operator T = L:r=l fi ® Ui, where each fi

is a strictly positive linear functional on X, Ui > 0 holds for each i, and L:r=l Ui is a strong unit, is a Krein operator. Let fJ, be a nonzero Borel measure on a Hausdorff compact topological space o and let T: C(O) ---+ C(O) be a positive kernel operator generated by a continuous strictly positive kernel. That is, there exists a continuous realvalued function T(t, s) on 0 x 0 satisfying T(t, s) > 0 for all (t, s) E S1 x 0 and

Tf(t) = 10 T(t, s)f(s) dfJ,(s)

for each f E C(O). (As usual, we use the same letter for the kernel and the operator.) Then T is a compact Krein operator. Let T: [a, b] x [a, b] ---+ [0, 00) be a continuous function such that T( t, s) > 0 holds for all Is - tl < E, where E > 0 is fixed. Then the kernel operator generated by T(t, s), i.e., the operator T: C[a, b] ---+ C[a, b] defined by

Tf(t) = lb T(t,s)f(s)ds,

is a compact Krein operator. To see this, note first that T2 is also a kernel operator with kernel given by the formula

T2(t,S) = lb T(t,u)T(u,s)du.

Now we claim that T2(t, s) > 0 holds for all Is - tl < 2Eo To see this, let s < t and t - s = Is - tl < 2Eo Let Uo = sit, the mid-point of the interval

[s, t], and put 8 = E - Is~tl. Clearly, 0 < 8 < E, and if [uo - ul < 8, then Is - tl Is - ul ::; Is - uol + lu - uol < -2- + 8 = E,

and similarly It - ul < E. Hence T(t,u) > 0 and T(u,s) > 0 for all u E (uo - 8, Uo + 8), and so

T2(t, s) = lb T(t, u)T(u, s) du ~ lUo+<, T(t, u)T(u, s) du > O. a uo-Ii

Repeating the above process, we see that there exists a positive integer n such that Tn(t, s) > 0 holds for all (t, s) E [a, b] x [a, b]. It follows that T is a compact Krein operator. This example is an important special case of the preceding example: If T:[a,b] x [a,b] ---+ [0,00) is a continuous function such that T(t,t) > 0 for all a ::; t ::; b, then the kernel operator generated by T on C[a, b] is a compact Krein operator. Consider a continuous function T: [a, b] x [a, b] ---+ [0,00) such that the partial derivatives crT / EJtn and anT / asn exist and are continuous for all n = 1, ... ,k. Also, assume that T(t, t) > 0 holds for all to Then the operator T: Ck[a, b] ---+ Ck[a, b], defined by

Tf(t) = lb T(t, s)f(s) ds,


is a Krein operator. If T: X --t X is a Krein operator and S: X --t X is an arbitrary positive operator, then S + T is a Krein operator.

Three basic properties of Krein operators are included in the following lemma.

LEMMA 5.2 1fT: X --t X is a Krein operator, then: 1) T is a continuous operator; 2) T carries units to units; and 3) Tn is a Krein operator for each n 2:: 1 (and so Tn # I for each n 2:: 1).

Proof 1) This is a consequence of Corollary 2.5. 2) Let u be a unit. Since T is a Krein operator, there exists a positive integer

k 2:: 1 such that v = Tku is again a unit. Next, choose some ry > 0 such that Tk-1u ::; ryu and note that v = Tku ::; ryTu holds. This shows that Tu is a unit.

3) This is an easy consequence of part 2). Q.E.D.

The reader should observe that the product of two commuting Krein operators is also a Krein operator. However, the product of two noncommuting Krein operators need not be a Krein operator. For instance, the positive matrices A =

( ~ ~) and B = (~ ~) both define Krein operators on R2 while their

product matrix AB = (; ~) does not define a Krein operator.

6. Eigenvalues of Krein Operators

In this section we shall study some basic properties of the eigenvalues of Krein operators. We start with a lemma.

LEMMA 6.1 If T: X --t X is a Krein operator and Xo > 0 is a positive eigenvector corresponding to an eigenvalue AO, then:

1) AO > 0; 2) Xo is a strong unit; and 3) the eigenspace corresponding to AO, i.e., the vector subspace

NT(AO) = {x EX: Tx = AOX}, is one-dimensional.

Proof Assume that Txo = AOXO with Xo > O. Since T is a Krein operator, there exists a positive integer n such that Tnxo is a strong unit. From Tnxo =

(>'o)nxo, it follows that Xo is a strong unit and that >'0 > 0. Now suppose that Tx = AOX holds for some x # O. By Lemma 3.3, there

exists an a E R such that Xo + ax E oX+; clearly, a i= O. If Xo + ax > 0, then from T(xo + ax) = AO(XO + ax) and in view of 2), we infer that Xo + ax is a strong unit. The latter (in view of Lemma 3.2) implies that Xo + ax E Int(X+),

which is impossible. Thus, Xo + ax = 0 or x = (-±) Xo, which shows that

NT(AO) is one-dimensional. Q.E.D.

14 Y.A. ABRAMOVICH, CD. ALIPRANTIS AND O. BURKINSHAW

And now we come to a fundamental eigenvalue property of Krein operators.

THEOREM 6.2 (M.G. Krein) A Krein operator T: X -+ X has (up to a scalar multiple) at most one positive eigenvector. Moreover, if Xo > 0 is the positive eigenvector ofT with corresponding eigenvalue '\0, then .\0 > 0 and every other real eigenvalue .\ of T satisfies 1.\1 ::; .\0.

Proof Assume that Xo > 0 is a positive eigenvector for T corresponding to the eigenvalue .\0. By Lemma 6.1, we know that .\0 > 0 and that Xo is a strong unit and (essentially) the only eigenvector corresponding to Ao.

Now suppose that .\ is a real eigenvalue of T satisfying 1.\1 > .\0. Pick a nonzero vector x such that Tx = .\x. Since Xo is a unit, there exists an a > 0 such that ±x ::; axo. Hence, m(±x) ::; aTnxo for each n, or .\n(±x) ::; a(.\otxo

for all n. It follows that I ~ In (±x) ::; axo for all n. Since I :0 I > 1 and X is Archimedean, we infer that ±x ::; 0 or x = 0 holds, which is a contradiction. Hence 1.\ I ::; .\0 holds, which shows that T does not have any real eigenvalue with absolute value greater than Ao.

If now Xl > 0 is an eigenvector corresponding to an eigenvalue '\1, then by Lemma 6.1 the eigenvector Xl is a strong unit and .\1 > 0. By the above discussion .\1 ::; .\0 and .\0 ::; .\1 both hold, and so .\1 = .\0. Consequently, by Lemma 6.1, 3), Xl is a scalar multiple of xo, and the proof is finished. Q.E.D.

The following theorem of M.G. Krein [8, Thm. 6.3] reveals an important eigenvalue property of adjoint operators.

THEOREM 6.3 (M.G. Krein) The adjoint of an arbitrary positive operator on a Krein space has a positive eigenvector corresponding to a nonnegative eigenvalue.

Proof Let T: X -+ X be a positive continuous operator on a Krein space X. By Lemma 5.2, 1) the operator T is automatically continuous. Fix some unit u and then pick some r > 0 such that Ilxll ::; r implies 1.£ ± X E X+. Thus, if f E X~ and X E X satisfies Ilxll ::; 1, then f(u ± rx) 2 0, or If(x)1 ::; f~U). Consequently, Ilfll ::; f~u) holds for each f E X~.

Next, consider the set

C = {f E X~ : f(u) = I}.

Clearly, C is nonempty, convex, and w* -closed. By the above, we also have Ilfll ::; ~ for each fEe, and so C is a norm bounded subset of X'. Hence, C is a nonempty, convex, and w* -compact subset of X'.

Next, define the mapping F: C -+ C by

f +T'f F(f) = [f+T'fl(u)

f +T'f 1 + T' f(u)'

(1)


A straightforward verification shows that F indeed maps C into C and that F: (C, w*) -r (C, w*) is a continuous function. Therefore, by Tychonoff's Fixed Point Theorem there exists a ¢ E C such that F(¢) = ¢. That is, ¢ + T'¢ =

[1 + T'¢(u)]¢, or T'¢ = [T'¢(u)]¢, which shows that 0 < ¢ E X~ is an eigenvector for T' corresponding to the nonnegative eigenvalue T'¢(u). Q.E.D.

A proof of Theorem 6.3 that does not use fixed point theorems can be found in [6]. Recall that a hyperinvariant subspace for a continuous operator T: X -r X on a Banach space is a closed subspace that is invariant under every continuous operator on X that commutes with T.

THEOREM 6.4 On a Krein space, every positive operator which is not a multiple of the identity has a nontrivial hyperinvariant subspace.

Proof Let X be a Krein space and let T: X -r X be a positive operator which is not a multiple of the identity. By Theorem 6.3 there exist a A 2: 0 and some 0 < ¢ E X, such that T'¢ = A¢. Put A = T - AI and note that A'¢ = O. It follows that Y = A(X) is a nontrivial closed subspace of X. To complete the proof, notice that if a continuous operator S: X -r X satisfies ST = TS, then S(Y) ~ Y holds. Q.E.D.

Since a Krein operator cannot be a multiple of the identity operator, we have the following consequence.

COROLLARY 6.5 Every Krein operator has a nontrivial hyperinvariant subspace.

The following result describes a large class of nonquasinilpotent operators.

THEOREM 6.6 Every Krein operator has a strictly positive spectral radius. Proof Let T: X -r X be a Krein operator. By Theorem 6.3 there exist a

positive eigenvector 0 < f E X, and a nonnegative scalar A such that T' f = AI. We claim that A > 0 (and hence r(T) = r(T') 2: A > 0 must hold).

Fix 0 < x E X+ and then pick some integer n such that u = Tnx is a strong unit. If A = 0, then

which is impossible. Hence, A > 0 must hold, as desired. Q.E.D.

A positive eigenvalue AO of a continuous operator on a Banach space is said to be a leading eigenvalue whenever AO 2: IAI holds for every other eigenvalue A of the operator.

COROLLARY 6.7 If T: X -r X is a compact Krein operator, then the spectral radius r(T) of T is a leading eigenvalue.


Proof By Theorem 6.6, we have r(T) > O. So, by the classical Krein-Rutman Theorem [8] (see also [5, Thm. 5.3.7, p. 182]), we know that the spectral radius r(T) is an eigenvalue and we are done. Q.E.D.

Corollary 6.7 applied to matrices yields the classical result of O. Perron [13] and G. Frobenius [4]; see also [18, Chapt. 19].

COROLLARY 6.8 (perron-Frobenius) If A is a nonnegative n x n matrix such that for some k ;:::: 1 the matrix A k has strictly positive entries, then the spectral radius of A is a strictly positive eigenvalue of multiplicity one having a strictly positive eigenvector.

7. Fixed Points and Eigenvectors

In this section we shall discuss several interrelationships between fixed points and eigenvectors of families of commuting positive operators on Krein spaces. Before stating the results, we shall clarify a crucial connection between fixed points and eigenvectors.

Let T: X --+ X be a positive operator on a Krein space, let U E X be a fixed strong unit and consider the nonempty, convex, and w* -compact set

C = {f E X~ : f(u) = I}.

Also, as in the proof of Theorem 6.3, we consider the continuous mapping F: (C, w*) --+ (C, w*) defined by

f +T'f F(f) = [f + T' fl(u)

f +T'f 1 + T' f(u)"

The important thing to emphasize here is that a linear functional ¢ E C is a fixed point for F if and only if ¢ is an eigenvector for the adjoint operator T' corresponding to the eigenvalue T'¢(u). Indeed, if F(¢) = ¢, then it is clear that T'¢ = [T'¢(u)]¢ holds. Conversely, if ¢ E C satisfies T'¢ = A¢, then

F( ) = ¢ + T' ¢ = ¢ + A¢ = ¢ [¢ + T'¢](u) 1 + A ¢.

The next result generalizes Theorem 6.3. (Keep in mind that by Lemma 5.2 every positive operator on a Krein space is automatically continuous.)

THEOREM 7.1 (M.G. Krein) Let {Ta}aEA be a family of pairwise commuting positive operators on a Krein space X. Then the family of adjoint operators {T~}aEA has a common positive eigenvector, i.e., there exist some 0 < ¢ E X' and a family of scalars {Aa } aEA (which is necessarily a family of nonnegative scalars) such that T~¢ = Aa¢ holds for each a.


Proof Fix a unit 11, in X and consider the nonempty, convex and w* -compact set

e = {J E X~: feu) = I}.

Also, for each index a consider the continuous mapping Fa: (e, w*) -+ (e, w*) defined by

f+T~f Fa(f) = [f + T~fl(11,)

f+T~f

1 + T~f(ur As noticed before, each fixed point of Fa is an eigenvector of T~ corresponding to a nonnegative eigenvalue. Let Da denote the set of all fixed points of Fa, i.e., let

Da = {f E e: Fa(f) = n· By Theorem 6.3 each Da is a nonempty and w* -compact subset of e. To complete the proof we must show that naEA Da # 0, or equivalently that the family of w*-compact sets {Da: a E A} has the finite intersection property.

The proof goes by induction. As mentioned above, each Da is nonempty. So, for the induction step, assume that every intersection of any n members of the family {Da : a E A} has a nonempty intersection and let 001, ... ,an, an+l be n + 1 arbitrary indices. We must show that n~l Dai # 0.

To this end we note that (by our induction hypothesis) there exists a 0 < $ E

Dal n ... n Dan' So, if Ai = T~i $( 11,) :2: 0, then it follows from Fai ($) = $ that T~i$ = Ai$ for each i = 1, ... , n. Now for each I ~ i ~ n we define the set

ei = {J E e : T~J = Ad}·

Clearly, each ei is convex, w* -compact, and (since $ E ei ) nonempty. Next we claim that Fan+l (ei ) ~ Ci holds for each 1 ~ i ~ n. Indeed, if fECi, then

I ( f) I ( f + T~n+J ) Tai Fan+l =Tai l+T~n+J(u)

T~i f + T~i T~n+l f 1 + T~n+J(u)

T~if + T~n+l T~if 1 + T/:"n+J( 11,)

Ad + AiT~n+J 1 + T~n+J(u)

( f +T~n+J ) = Ai 1 + T~n+J(u) = Ai (Fan+l I),

18 YA. ABRAMOVICH. C.D. ALIPRANTIS AND O. BURKINS HAW

which shows that Fan+J E C. Now if we let G = n~l Gi , then G is a convex, w*-compact, and (in view of

rP E G) nonempty subset of G such that Fan+l (G) <;: G. By Tychonoff's Fixed Point Theorem there exists a 0 < 1/J E G with Fan +l (1/J) = 1/J. Since 1/J E Gi, we

have T~i 1/J = Ai1/J, and hence Fai (1/J) = 1/J for each 1 ~ i ~ n. So, 1/J E nr~/ Dai and hence n~i/ Dai Ie 0. The proof of the theorem is now complete. Q.E.D.

Theorem 7.1 was proved by M.G. Krein under the extra hypothesis that Ta(U) <;: U holds for each a. Our Theorem 7.1 implies, of course, this version of M.G. Krein. However, it is interesting to know that M.G. Krein's version also implies our Theorem 7.1. To see this, let Ta,E = Ta + EI for each 0 < E < 1, where I: X --t X is the identity operator. Clearly, Ta,E(U) c U holds, and so for each E > 0 there exist a 0 < rPE E X, and a family of nonnegative scalars {Aa,E}aEA such that

(2)

The net {rPE}O<E<1 (where EI ~ E2 means El ~ E2) in G has a w*-convergent

subnet. So, by passing to a subnet if necessary, we can assume that rPE ~ rP holds in G. Now fix a E A and then pick some t > 0 with TaU ~ tu. From (2) it follows that

o ~ Aa,E = Aa,ErPE(U) = T~rPE(U) + ErPE(U) ~ t + 1.

This implies that the net {Aa,E}O<E<l has an accumulation point Aa 2': O. Now a glance at (2) guarantees that T~rP = AarP.

The fixed point analog of Theorem 7.1 is as follows.

THEOREM 7.2 (M.G. Krein) Let {Ta}aEA be a family of pairwise commuting positive operators on a Krein space X. If there exists a strong unit U E X such that Ta( u) = U for all a, then there exists a nonzero positive linear functional f E X' such that T~f = f for all a.

Proof Assume that for some strong unit U we have Ta(u) = U for each a E A. Since -tu ~ x ~ tu implies -tTa(U) = tu ~ Ta(x) ~ tTa(U) = tu, it follows that IITa(x) Ilu ~ Ilxllu holds for each x E X and each a E A. In particular, if k l , . .. ,kn are arbitrary nonnegative integers and a I, ... ,an are arbitrary indices, then

IIT~:T~~ ... T~~xllu ~ Ilxllu (**)

holds for each x E X. Next, we consider the subspace Y of X defined as follows:

Y = { x EX: :3 al, ... ,an E A and XI, .. . ,Xn EX

with x = t,(TatXi - Xi) } .


We claim that Y n U = 0. To see this, assume by way of contradiction that Y n U Ie 0. So, there exist indices al,· .. ,an and vectors XI, ... ,Xn such that the vector e :Li=1 (Taixi - Xi) is a unit. Now fix m > n and define the operator

where ai, ... , an are the fixed indices appearing in the representation of e as an element of Y. Clearly, T m is a positive operator and T m ( u) = u holds. Since the family {Ta}aEA consists of pairwise commuting operators, we see that

If we let c = max{llxillu: i = 1, ... , n}, then from (**) it follows that

II T~~ (T;:+l - TaJXi #i u

:::; 2e.

Consequently, from the triangle inequality we infer that

1 2rn IITm(e)llu:::; -nmn - 12e =-. mn m

Now, note that if TJ > 0 satisfies u :::; TJe, then

2TJrn 1 = Ilullu = IITm(u)llu :::; TJIITm(e)llu :::; -

m

holds for each m > n, which is impossible. Thus, indeed Y n U = 0. Finally, since U = lnt( X +), there exists (by the classical separation theorem)

a nonzero f E X, such that f(y) = 0 for all y E Y and f(x) 2: 0 for all X E U. Taking into account that U = X+, we conclude that f > O. From

(X, T~f - f) = (TaX - X, f) = 0,

we see that T~f = f for each a E A, i.e., f > 0 is a fixed point for the family of operators {T~} aEA, and the proof is finished. Q.E.D.

20 Y.A. ABRAMOVICH, CD. ALIPRANTIS AND O. BURKINS HAW

COROLLARY 7.3 Let T: X ---J- X be a positive operator on a Krein space. If a nonzero real number)' is an eigenvalue of T having an eigenvector which is a unit, then ). > 0 and the adjoint operator T' has also ). as an eigenvalue.

The next result is a famous theorem of A.A. Markov [9] and is a surprisingly simple consequence of Theorem 7.2. In order to formulate it we need some preliminary discussion.

Recall that if S an arbitrary nonempty set, then £00 (S) denotes the Banach lattice of all bounded real valued functions on S. With the sup norm £00 (S) is an AM-space with unit, and it is in particular a Krein space. Now if ¢: S ---J- S is a mapping, then ¢ defines a positive continuous operator T¢:£oo(S) ---J- £oo(S) via the formula

T¢(x)(s) = x(¢(s)), x E foo(S), s E S.

Also, recall that the dual £'00 (S) can be identified with the space of all finitely additive measures on S (more precisely on the power set of S).

THEOREM 7.4 (Markov) Let {¢a}aEA be afamily ofpairwise commuting mappings on an arbitrary nonempty set S into itself, and let Ta = T¢cx for each a. Then there exists a positive functional 0 < f E f'oo(S) such that

f(Tax) = f(x)

for all a and all x E £oo(S). Proof Notice again that foo(S) is a Krein space, Also, the family {Ta}aEA

consists of pairwise commuting positive operators on foo (S). In addition, if 1 denotes the constant function one on S, then 1 is a unit of foo(S) and Ta(l) = 1 holds for each a. The conclusion now is a direct consequence of Theorem 7.2, Q.E.D.

A companion theorem to Theorem 7.2 is the following result.

THEOREM 7.5 Let {Ta} be a family of pairwise commuting positive contractions on a Krein space X. If there exists a nonzero vector v E X such that Ta( v) = v holds for each a, then there exists a nonzero positive linear functional f E X' satisfying T~f = f for each a.

Proof Start by fixing some nonzero vector v E X such that Ta ( v) = v holds for each a. By Theorem 3.4 there is a unit u such that Ilxll :s; 1 implies -u :s; x :s; u (and hence Ilxll u :s; 1). In particular, since Ilxll :s; 1 implies IITa(x)11 ::; IITal1 . Ilxll :s; Ilxll ::; 1, we see that IITa(x)llu ::; 1 holds for each Ilxll ::; 1.

Next, consider the positive operators Ta: (X, II . II) ---J- (X, II . Ilu) and notice that IITallu = sUPllxI191ITa(x)llu ::; 1. Consequently,

IITa(x)llu::; IITallu' Ilxll ::; Ilxll


holds for each x E X. Moreover, note that if k1' k2 , . .. , kn are arbitrary positive integers and 001,002, ... , an arbitrary indices, then

IITkITk2 ... Tknxll = liT (Tkl- 1T k2 ... Tknx)11 al a2 an u al al a2 an u

< IITkl- 1T k2 ..• Tknxll - al a2 an

~ Ilxll·

The rest of the arguments are as in the proof of Theorem 7.2. We consider the same vector subspace Y. Assume that Y n U f= 0, and then for each m > n construct the operator Tm. If we let e = max{llxill: i = 1, ... , n}, then it follows that

2en IITm(e)llu ~ -.

m Now if "7 > a satisfies ±v ~ "7e, then

0< Ilvllu = IITa(v)llu ~ "7II Ta(e)llu ~ 2"7cn m

holds for each m > n, which is impossible. The rest of the proof is exactly as the last part of Theorem 7.2. Q.E.D.

It should be noted that Theorem 7.5 does not follow from Theorem 7.2. For instance, if X = R2 and T: X -+ X is the positive projection represented by the

matrix (~ ~ ), then T has 1 as an eigenvalue with eigenvector ( ~ ). However,

T does not have any strong unit as an eigenvector.

COROLLARY 7.6 If a positive contraction T: X -+ X on a Krein space has 1 as an eigenvalue, then there exists an a < f E X' such that T' f = f.

Finally, the following example shows that the commutativity of the family of operators in Theorems 7.2 and 7.5 cannot be dropped.

EXAMPLE 7.7 Let P denote the family of all (Borel) probability measures on [0,1]. For each J1 E P consider the positive operator TJ.L: e[O, 1] -+ e[o, 1]

defined by TJ.Lf(t) = Jd f(s) dJ1(s), i.e., TJ.L = J1 ® 1, where 1 denotes the constant function one on [0, 1]. Clearly, TJ.LTv = Tv holds for all J1, v E P. In particular, it follows that {TJ.L : J1 E P} is a family of noncommuting positive projections on e[O, 1]. In addition, notice that TJL1 = 1 for each J1 E P.

Assume now by way of contradiction that there exists a ° < 8 E M[O, 1] such that T~8 = 8 holds for each J1; we may assume that 8([0,1]) = 1. Hence,

e = (T~)e = (1 ® J1)e = (1, e)J1 = J1

holds for each J1 E P, which is impossible. Consequently, the family {T~: J1 E P} does not have any common fixed point.

Finally, we mention that several results presented in sections 6 and 7 have been generalized by LA. Bakhtin in [1], [2].


Acknowledgement

The authors thank Professor E.M. Semenov of Voronezh State University for reading the manuscript and providing useful comments.

References

1. LA. Bakhtin, On the existence of eigenvectors of linear positive noncompact operators, Math. USSR-Sh. (N.S.) 64 (1964), pp. 102-114 (in Russian).

2. LA. Bakhtin, On the existence of common eigenvector for a commutative family of linear positive operators, Math. USSR-Sh. (N.S.) 67 (1965), pp. 267-278 (in Russian).

3. LA. Bakhtin, M.A. Krasnoselsky and v.Ya. Stezenko, Continuity of linear positive operators, Sihirsk. Math. Z. 3 (1962), pp. 156-160.

4. G. Frobenius, Uber Matrizen aus nicht-negativen Elementen, Silz. Berichte Kgl. Preu(3. Akad. Wiss. Berlin (1912), pp.456-477.

5. G. Jameson, Ordered Linear Spaces, Lecture Notes in Mathematics, 141, Springer-Verlag, Heidelberg, 1970.

6. A.K. Kitover, The spectral properties of weighted homomorphisms in algebras of continuous functions and their applications, Zap. Nauen. Sem. Leningrad Otdel. Mat. Inst. Steklov. (WMl) 107 (1982), pp. 89-103.

7. M.G. Krein, Fundamental properties of normal conical sets in a Banach space, Dokl. Akad. Nauk USSR 28 (1940), pp. 13-17.

8. M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk 3 (1948), pp. 3-95 (in Russian). Also, Amer. Math. Soc. Trans!. 26 (1950).

9. A.A. Markov, Some theorems on Abelian collections, Dokl. Akad. Nauk USSR 10 (1936), pp. 311-313.

10. L. Nachbin, On the continuity of positive linear transformations, Proc. Internat. Congress of Math. (1950), pp. 464-465.

11. 1. Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957). 12. A.L. Peressini, Ordered Topological Vector Spaces, Harper & Row, New York, 1967. 13. O. Perron, Zur Theorie der Matrizen, Math. Ann. 64 (1907), pp.248-263. 14. H.H. Schaefer, Topological Vector Spaces, Springer-Verlag, Berlin-New York, 1971. 15. B.z. Vulikh, Introduction in the Theory of Cones in Normed Spaces, Kalinin State Uni

versity, 1977. 16. B.Z. Vulikh, Special Topics in Geometry of Cones in Normed Spaces, Kalinin State Uni

versity, 1978. 17. Y.c. Wong and K. F. Ng, Partially Ordered Topological Vector Spaces, Clarendon Press,

Oxford, 1973. 18. A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983.


23

A Remark on the Representation of Vector Lattices as Spaces of Continuous Real-Valued Functions

YURI A. ABRAMOVICH Department of Mathematics, IUPUI, Indianapolis, IN 46205-2810, U.S.A

and

WOLFGANG FILTER Mathematik, ETH-Zentrum, CH-B092 Zurich, Switzerland


Abstract. The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space 0, unique up to a hO!lleomorphism, such that L can be represented isomorphically as an order dense vector sublattice L of the universally complete vector lattice Coo(O) of all extended-real-valued continuous functions/on 0 for which {w E 0: Ifew)1 = co} is nowhere dense. Sin~e the early days of using this representation it has been important to fInd conditions on L such that L consists of bounded functions only. The aim of this short article is to present a simple complete characterization of such vector lattices.

Mathematics Subject Classifications (1991): 46A40, 46E05

Key words: representation by bounded functions, Archimedean vector lattice

The well-known Ogasawara-Maeda-Vulikh representation theorem discovered by Ogasawara-Maeda ([3], [5]) and independently by Vulikh [7] (and very similar to Nakano's representation theorem [4]), asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space n, (called the Stone space of n), unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice L of the universally complete vector lattice Coo(n) of all extended-real-valued continuous functions f on n for which {w En: If (w ) I = oo} is nowhere dense. Since the early days of using this representation it has been important to fmd conditions on L such that L consists of bounded functions only. Partial results in this direction were obtained by Ogasawara-Maeda, Papert [6] and Bemau [1] (see below). The aim of this short article is to present a simple complete characterization of such vector lattices.

Ogasawara-Maeda gave a partial solution of the problem. They considered Dedekind complete vector lattices L with the following property: If (Kn) is a sequence of upper bounded subsets of L such that the sequence (sup Kn) order converges in L, then each Kn contains a finite subset K~ such that (supK~) order con-

24 YURI A. ABRAMOVICH AND WOLFGANG FIL1ER

verges to the same limit. Then, in order that a vector lattice L of this kind can be represented such that L C C(O) holds, it is necessary and sufficient, according to Ogasawara-Maeda, that there exists a maximal disjoint system (UJI-EI in L+ such that for each vEL + there are a countable subset J of I and a bounded

family (al-)l-EJ of reals with v = LLEJ aLUl-. Later Papert has shown [6, Thm.7] that a vector lattice L can be represented

as some vector lattice L of real-valued functions iff for each 0 < U E L there exists, for each vEL, a natural number n( v), such that for any finite subset { VI, •.. , vp } of L,

1 sup --Vi i:. U

iE{I, ... ,p} neVi)

holds. Bernau [1, Thm.7] strengthened this result as follows: He proved that under the above condition L can be represented as an order dense vector sublattice of C(O'), where Of is some extremally disconnected locally compact Hausdorff space (in fact, 0' can be chosen as a dense open subset of the Stone space 0). Observe however, that the condition of Papert-Bernau is satisfied, for example, in each space Coo(O), where 0 is an arbitrary extremally disconnected compact Hausdorff space with a dense subset of isolated points, which shows that their condition fails to guarantee the existence of a nonextended-valued representation on the Stone space O.

In the following, let L be an Archimedean vector lattice, and let 0 be its Stone space. We say that

L satisfies (*) iff there exists a maximal disjoint system (Ul-)l-EI in L+ such that for each vEL + there exists n E N satisfying z ::; nUl- for each 0::; Z E {uL}dd, Z ::; v, and for each /., E I.

PROPOSITION 0.1 The following assertions hold. a) If L satisfies (*). then L can be represented as an order dense vector sub

lattice L of C(O). b) If L may be represented as an order dense vector sublattice L of C(O) and

if L is Dedekind complete, then (*) holds in L. Proof a) Since the Ul- are pairwise disjoint we can find a representation of L

on 0 such that each U L is mapped onto the characteristic function 10. of a clopen (closed and open) subset Ol- of 0, the OL being pairwise disjoint and U OL dense in O. Then, if v E L+ and n E N is given by (*), we have v ::; nl\l.

b) If L is Dedekind complete, by Zorn's Lemma we can then find a disjoint family (OL) of clopen subsets of n such that 10. E L for each i and U OL is dense in O. Letting U L be the element of L corresponding to 10, we obtain a system (Ul-) satisfying (*). Q.E.D.

The following simple example shows that (*) need not be satisfied if we drop the assumption of Dedekind completeness in b).

REPRESENTATION OF VECTOR LATTICES 25

EXAMPLE 0.2 Let L := {J E C([O, 1]): 1(0) = O}. Since L is obviously an order dense vector sublattice of C([O, 1]), it can be represented as an order dense vector sublattice of C(O) (where 0 is the Stone space of C([O, 1])). But if 1 E L+ \ {O} one can easily construct 9 E {J}dd, 9 2: 0, such that -7 is (defined and) unbounded: Indeed, there is a sequence (In)n>rzo of pairwise disjoint nonempty open intervals in [0, 1] such that :;& ~ 1 ~ £ on In. Let gn

be an element of C([O, 1]) such that gn == ° on [0, 1]\In and maxtE[O,lj gn(t) = ~. Set g:= L.n2rzo gn. Since 9 1:. nl for all n E N, and 1 was arbitrary, (*) cannot be satisfied in L.

Let us now formulate a condition which (though slightly cumbersome) turns out to be appropriate for the problem under consideration: We say that

L satisfies (**) iff there exists a family (U~A)~EI,AEA, in L+ such that

1) L = {U~A : tEl, >. E A~}dd; 2) for each tEl, the net (U~A)AEA, increases and is order bounded in L,

i.e., ° ~ U~A h ~ x~; 3) for all tl, t2 E I, tl i- t2, and for all Al E A~i' A2 E A~2 we have

inf( U~l,A" U~2,AJ = 0; 4) for each v E L+ there exists n E N satisfying SUPAEA, inf(z, nU~A) = z

for each ° ~ z E {U~A: >. E A~}dd, z ~ v, and for each tEl.

THEOREM 0.3 L can be represented as an order dense vector sublattice of C(O) if and only if L satisfies the condition (**).

Proof Denote by L8 the Dedekind completion of L. First we show the sufficiency of (**). Set U~ := SUPAEA, U~A in L8. Then

(U~)LEI obviously satisfies (*) for each v E L+, hence also for each v E (L8)+. Thus, by a) of our proposition, L8 (and therefore L as well) can be represented in the desired way.

Now let L be embedded order densely in C(O). Then, by [2, 50.8(ii)], the same holds for L8. By part b) of our proposition, there exists a maximal disjoint system (UL)~EI in (L8)+ satisfying (*). For each tEl, there is a net (ULAhEA, in L+ with ° ~ UtA h ~ U~. Obviously, (UtA)tEI,AEA, satisfies (**). Q.E.D.

In conclusion we remark for the sake of completeness that our condition (**) implies the conditions of Papert-Bemau and of Ogasawara-Maeda (and this is quite natural, since the representation on the Stone space is, in a way, the best possible, and therefore, hardest to obtain). Indeed, in view of our theorem, it is sufficient to verify that their conditions are satisfied in an arbitrary order dense vector sublattice L of C(O), where 0 is an extremally disconnected compact Hausdorff space.

If U E L, U > 0, then choose n(v) 2: 211vll/lluII (where II . II denotes the supremum norm) to see that the Papert-Bemau condition holds. Now let L satisfy

26 YURI A. ABRAMOVICH AND WOLFGANG FILlER

additionally the (rather restrictive) assumptions of Ogasawara-Maeda. Then each elopen U c n with lu E L must be finite; therefore, by the argument used in part b) of our proposition, n is the Stone-Cech compactification of some discrete space n' c n, and it is easy to see that for each U E L the set {w E n: u( w) f O} nn' must be countable. This implies the Ogasawara-Maeda condition.

References

1. S.J. Bemau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math Soc. 15 (1965), pp. 599-631.

2. W.A.J. Luxemburg, A.C. Zaanen, Riesz spaces /, North-Holland, Amsterdam-London, 1971.

3. F. Maeda, T. Ogasawara, Representation of vector lattices, J. Sci. Hiroshima Univ. Ser. A 12 (1942), pp. 17-35 (in Japanese).

4. H. Nakano, Eine Spektraltheorie, Pmc. Phys.-Math. Soc. Japan (3) 23 (1941), pp. 485-511.

5. T. Ogasawara, Remarks on the representation of vector lattices, J. Sci. Hiroshima Univ. Ser. A 12 (1943), pp. 217-234 (in Japanese).

6. D. Papert, A representation theory for lattice groups, Proc. London Math. Soc. (3) 12 (1962), pp. 100-120.

7. B.z. Vulikh, Concrete representations of linear partially ordered spaces, Dokl. Akad. Nauk SSSR 58 (1947), pp. 733-736 (in Russian).


27

Domination of Uniformly Continuous Semigroups

W. ARENDT Equipe de Mathematiques, Universiti de Franche-Comte, 25030 Besam;on Cedex, France

and

J. VOIGT Fachbereich Mathematik der Universitiit, 2900 Oldenburg, Germany

(Reveived: 27 April 1992)

Abstract. We prove that a bounded operator on a Banach lattice, satisfying a growth condition, is regular. Also, we prove that the generator of a Co-semigroup on such a lattice for which such an operator exists is bounded.

Mathematics Subject Classifications (1991): 47D03, 46A40

Key words: domination of operators, uniformly continuous seroigroups

Introduction

The aim of this note is to prove the following theorem.

THEOREM 0.1 Let T = (T(t))f?:,O be a Co-semigroup on a real or complex Banach lattice E. Assume that B E L(E) is a bounded operator such that

(t 2: 0, x E E). (1)

Then B is a regular operator. Moreover, the generator A of T is bounded.

Here a bounded operator B is called regular if B is dominated by a positive operator C E L(E) (i.e. IBxl :s; Clxl (x E E). If E is order complete, then every regular operator B possesses a modulus; i.e. there exists a smallest positive operator IBI dominating B (see Schaefer [9]).

In analogy one may ask whether a given Co-semigroup S which is dominated by a positive Co-semigroup possesses a modulus semigroup, i.e. a smallest positive semigroup S# dominating S. This question is treated by Derndinger [6], Becker-Greiner [4] and Charissiadis [5] (see also Nagel [7, C-II]). With the help of a result of Derndinger [6] (see also [7, C-II, Thm. 4.17]) from the above theorem one obtains a positive answer for uniformly continuous semigroups.

28 w. ARENDT AND J. VOIGT

COROLLARY 0.2 Assume that E is order complete. Let B E £(E) and assume that there exists a Co-semigroup T satisfying (1).

Then

(t 2: 0, x E E), (2)

where B# = ReBo+IBll with Bo E Z(E), Bl E Z(E).l.. such that B = Bo+Bl.

Here Z(E) denotes the center of E (see Zaanen [11] or Aliprantis-Burkinshaw [1] where Z(E) is denoted by Orth(E)). Note that Z(E) is a band in C(E) so

that the decomposition of B is unique. The proof ofthe theorem is based on perturbation arguments (cf. [2]). For fur

ther relations between perturbation and domination we refer to [3] and Rhandi [8].

The Proofs

Let A be the generator of a Co-semigroup T = (T(t)h?o on a Banach lattice E. Denote by w(A) the type of T. Then (w(A), (0) c p(A). If T is positive, then

whenever w(A) < A < JL. (3)

Conversely, if (). - A)-l 2: 0 for all ). 2: w and some w > w(A), then T is positive. For a > 0 the operator aA generates the Co-semigroup Ta given by Ta(t) = T(at). Note that Ta is positive if and only if T is positive.

The following is an easy consequence of the Trotter -Kato theorem.

LEMMA 0.3 Let Bn E £(E) (n E N U {oo}) such that limn-too Bn = Boo in the operator norm. Denote by Sn the semigroup generated by A + Bn. Then limn-too Sn(t) = Soo(t) strongly for all t 2: o.

We will use the following result due to Derndinger [6] (see also [7, C-II, Lemma 4.18]).

PROPOSITION 0.4 Let A be the generator of a positive Co-semigroup on E. If A 2: 0 (i.e. Ax 2: 0 for all 0 :s; x E D(A)), then A is bounded.

The following generalization of Proposition 0.4 is the key step in the proof of the theorem.

PROPOSITION 0.5 Let A be the generator of a positive semigroup and let B E

£( E) be a real bounded operator. If A - B 2: 0, then A is bounded and therefore regular(cf [7, C-II, Thm. 1.11], [10]).

Proof By Proposition 0.4 it suffices to show that the semigroup generated by A-B is positive. For a 2: 0 the operator Aa = A+a(A-B) = (1 +a)A-aB generates a Co-semigroup Sa. Let M = {a 2: 0: So. is positive }. We claim that M = [0, (0).

DOMINATION OF UNIFORMLY CONTINUOUS SEMIGROUPS 29

a) Let a E M. Then there exists E > 0 such that [a, a + E) C M. In fact, let A > w(Ac,,). Let C = A - B and c = IICR(A,AaJII- l . Then by (3), for 0 < b < E and /L 2: A, IlbCR(/L, Aa)11 < 1 and so (I -bCR(/L,Aa))-l = "L::=o(bGR(/L, Aa))n 2: 0 (where R(/L, Aa) = (/L-Aa)-l). Since (/L-AaH) = (/L - Aa - bC) = (I - bC(/L - Aa)-l)(/L - Aa), it follows that R(/L, AaH) = R(/L, Aa)(I - bCR(/L,Aa))-l 2: 0 for all /L 2: A, 8 E (O,c). Consequently a+b E M for 8 E (0, c).

b) Assume that 'Y := sup { a 2: 0: [0, a) C M} < 00. Note that for a 2: 0 the semigroup Sa is positive if and only if the semigroup generated by 1 ~a Aa = A - l~aB is positive. Letting a i 'Y it follows from Lemma 0.3 that S, is positive. Now a) leads to a contradiction. We have shown that Aa = (1 + a)AaB generates a positive sernigroup for all a 2: O. Consequently, the sernigroup generated by A - l~aB = l~aAa is positive as well. Letting a -+ 00 it follows from Lemma 0.3 that A - B generates a positive sernigroup. Q.E.D.

Proof of the Theorem. Since letBxl :S T(t)x (x E E+) by assumption, it follows that

(R B) 1· Re(etBx) - x 1· T(t)x - x A e x = 1m < 1m = x

tlO t - tlO t

for all 0 :S x E D(A)+. Hence A - ReB 2: O. This finishes the proof in the real case. If E is complex we merely conclude that Re B is regular. In order to show that 1m B is regular we first assume that E is order complete. Then by [10] the band projection from creE) onto Z(E) has a contractive extension P: creE) -+ Z(E) (where the closure is understood in £(E)). Since IMI:S IIMIII for all M E Z(E) one has

IPCI :S IIGIII (G E .cr(E)). (4)

Denote by :1 the identity mapping on .c(E). Since etB E £T(E) (t > 0), it follows that B E creE). We already know that A is bounded. Since :1 - P is positive on creE), it follows from (1) that

1(:1 - P)etBI :S (:1 - P)letBI :S (:1 - P)etA (t 2: 0).

Since (:1 - P)I = 0, it follows for x E E+ that

1(:1 - P)Bxl = lim C 11(:1 - P)etBxl t->O

~ lim c 1(J - P)etAx t-.o

= (:1 - P)Ax :S (A + IIAII)x.

Hence B - PB is regular, and so B is regular. Moreover, (:1 - P)(ImB) =

Im((:1 - P)B) :S A + IIAIlI. Since by (4) P(lmB) :S II ImBllI :S IIBIII, it follows that

1mB :S A + (IIAII + IIBII)I. (5)

30 W. ARENDT AND J. VOIGT

Now assume that E is arbitrary. Since A is bounded and E' is order complete, applying the preceeding to A' and B' one obtains

1mB' ::; A' + (lIA'11 + IIB'II)!·

Hence (5) holds for A and B as well. Consequently, 1m B is regular. Q.E.D.

REMARK 0.6 By sligth modifications of the proof one obtains the following more general result. Let A be a densely defined operator such that for some W E R, [w, (0) c p(A) and

sup{IIAR()., A) II : A 2: w} < 00.

Assume that B E £(E) such that [w, 00) C pCB) and

R(A, B)x ::; R(A, A)x (A2:w, xEE).

Then B is regular and A is bounded.

We conclude with pointing out two open questions. In the first we formulate a generalization of the theorem presented above.

QUESTION 1 Let B E £( E) and assume that etB is regular for all t 2: O. Does it follow that B is regular?

The second concerns the possibility of generalizing Proposition 0.4 and 0.5.

QUESTION 2 Let A be the generator of a Co-semigroup S. Assume that a) D(A)+ is dense in E+; and b) Ax 2: 0 for all x E D(A)+.

Does it follow that T is positive?

If A generates a positive Co-semigroup, then a) is satisfied. In fact, if x E E+, then x = limA-+oo AR(A, A)x E D(A)+.

However, as the following example shows, there exists a generator B of a Co-semigroup such that D(B)+ = {O}. Thus condition b) is trivially satisfied and does not imply positivity without additional conditions.

EXAMPLE 0.7 Let E = V(O, 1) (1 ::; p < 00) and let A be the generator of the right shift semigroup, i.e. A is given by D(A) = {f: [0,1] -+ R absolutely continuous, f' E E, f(l) = O}, Af = -f'. Let n c (0,1) be measurable such that meas(n n (a, b)) > 0 and meas(nC n (a, b)) > a for all intervals o i= (a, b) C (0,1). Let :J: E -+ E be given by :J f = 10f - locf. Then :J is an isometric isomorphism and :J- 1 = :J. Let B be the generator of the semigroup T = (:JU(t):J)t?o. Then D(B)+ = {O}. In fact, let 0 ::; f E D(B). Then :J f E D(A). Thus :J f is continuous and :J f 2: 0 on n and ::; 0 on nc. Consequently, :J f = O.

DOMINATION OF UNIFORMLY CONTINUOUS SEMIGROUPS 31

Acknowledgement

The authors are grateful to H. Raubenheimer for several stimulating discussions on this subject.

References

1. C. Aliprantis, O. Burkinshaw, Positive operators, Acad. Press, London, 1985. 2. W. Arendt, Resolvent positive operators and integrated semigroups, Semesterbericht Funk

tiolWlanalysis Tiibingen Band 6 (1984), pp. 73-101. 3. W. Arendt, A. Rhandi, Perturbation of positive semigroups, Archiv der Mathematik, to

appear. 4. 1. Becker, G. Greiner, On the modulus of one-parameter semigroups, Semigroup Forum

34 (1986), pp. 185-201. 5. P. Charissiadis, On the modulus of semigroups generated by operator matrices, Semester

bericht FunktiolWlanalysis Tiibingen Band 17 (1989/90), pp. 1-9. 6. R. Derndinger, Betragshalbgruppen normstetiger Operator-halbgruppen, Arch. Math. 42

(1984), pp. 371-375. 7. R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Mathe

matics 1184, Springer, Berlin, 1986. 8. A. Rhandi, Perturbations positives des equations d'evolution et applications, These. Be

sanr,:on (1990). 9. H.H. Schaeffer, Banach lattices and positive operators, Springer, Berlin, 1974.

10. J. Voigt, The projection onto the center of operators in a Banach lattice, Math. Z. 199 (1988), pp. 115-117.

11. A.C. Zaanen, Riesz spaces II, North-Holland, Amsterdam, 1983.


Sums and Extensions of Vector Lattice Homomorphisms

SoJ. BERNAU Department of Mathematical Sciences, The University of Texas at El Paso, EI Paso, Texas 79968-0514


33

Abstract. This paper gives a summary account, with minimal or no proofs, first of some results which characterize order bounded linear operators which are sums of lattice homomorphisms, or more generally of orthomorphisms; and secondly of theorems concerning extensions of vector lattice homomorphisms (theorems of Hahn-Banach type if you will). In all cases we assume that domain and range are vector lattices and that the range is Dedekind complete. The results vary from historical (pre-1940) to recent (1990). The most recent work, on sums of lattice homomorphisms, is covered in §1 and the more classical work on extension theorems is dealt with in §2.

Mathematics Subject Classifications (1991): 47B60, 47A20, 47B65

Key words: order bounded operator, orthomorphism, vector lattice homomorphism, Hahn-Banach type theorems

Introduction

The results discussed in this paper have all been published or accepted for publication, accordingly we omit most proofs and many details of the proofs we do describe. We assume throughout that E and F are real vector lattices. For unexplained terminology we refer the reader to [1], [17], [19], [21].

1. Sums of Disjointness Preserving Operators

A necessary and sufficient condition for a linear transformation between vector lattices to be a finite sum of lattice homomorphisms.

Joint work of the author, c.B. Huijsmans and B. de Pagter.

DEFINITION 1.1 Let n be a positive integer and T a linear operator from E to F. We say that T is n-disjointifT is order bounded and for all Xo, Xl,. 0 0 ,Xn E E such that IXilA IXjl = 0 for all i i= j we have /\~o ITxil = o.

As we have discussed in [4], every I-disjoint positive linear operator from E to F is a Riesz homomorphism, and a general I-disjoint linear operator is order

34 SJ. BERNAU

bounded and disjointness preserving (i.e., if lul/\ Ivl = 0, then ITul/\ ITvl = 0). We can readily check that every (positive) linear operator on Rn is the sum of n (Riesz homomorphisms) disjointness preserving operators. To see this take the matrix representation relative to the standard basis of Rn (or any positive pairwise disjoint basis if greater generality is wanted) and write the matrix as the sum of n matrices each with at most one nonzero column.

We first recall some facts about disjointness preserving operators. These can be found, in [2], [7] or [18]. Note, however, that in [2] the term 'disjointness preserving' has a different meaning and that [2] considers selfmaps of E rather than maps from E to F. The relevant proofs in [2] all take place in the range of the operator and apply without change.

THEOREM 1.2 ([2], [7], [18]) Let E and F be Riesz spaces, with F Archimedean and T an order bounded linear operator from E into F such that ITul/\ ITvl = 0 for all u, vEE with lul/\ Ivl = O. Then there exist lattice homomorphisms T+, T-, and ITI from E to F such that T = T+ - T-, (T+)x = (Tx)+ and (T-)x = (Tx)- (0:::; x E E), ITI = T+ + T-, and ITxl = ITI(lxl) (x E E).

PROPOSITION 1.3 ([4, Prop. 2]) Let F be Archimedean, and suppose that the linear operators T1, ... , Tn are order bounded and disjointness preserving from E to F; if T = Lbl Ti, then T is n-disjoint. The Archimedean assumption on F can be dropped if T1 , ... ,Tn are assumed to be positive (and hence lattice homomorphisms ).

Proof Suppose Xo,· .. , Xn E E, and IXil/\ IXjl = ° for all i :I j. Then

n

1\ ITxil :::; L ITioXol/\·· . /\ ITinXnl, i=O

where the summation is over all choices of io, ... ,in from {I, ... , n}. In each summand we have at least two of the subscripts io, ... , in identical. Suppose we have k :ll and ik = il = j, then

llioxol/\ .. ·ITinXnl :S ITjXkl/\ ITjxzI = (ITjl(lxkl) /\ ITjl(lxzI)

= ITjl(lxlk /\ IxzI) = 0.

It follows that Ai=olTxil = 0, as required. The Archimedean property of F is used only for the existence and properties of the ITjl, so need not be assumed when the Tj are all lattice homomorphisms. Q.E.D.

We now consider sufficiency. In this connection we note that Huijsmans and de Pagter, [8, Remark 2.3], ask if a 2-disjoint positive operator is the sum of two lattice homomorphisms, and state that this property characterizes such operators if F = C(X), with X extremally disconnected. We need the standard description

VECTOR LATTICE HOMOMORPHISMS 35

of the minimal positive extension of the restriction of T to the solid subspace of E generated by u, [21, Thm. 83.8].

LEMMA 1.4 Let E and F be vector lattices with F Dedekind complete. Suppose that T is a positive linear map from E into F and 0 :::; U E E. If Tu: E -t F is defined by

00

Tux = V T(x 1\ nu) for 0:::; x E E (1) n=l

and Tux = Tu(x+) - Tu(x-) for arbitrary x E E, then Tu is linear from E to F. In the case that E = F and T is the identity, then Tu is the band projection on udd.

The remainder of §1 follows [4] very closely, so we give no further references.

LEMMA 1.5 Let E and F be vector lattices with F Dedekind complete, and T an n-disjoint positive linear operator from E to F. Suppose Uo, uo, ... ,Un E E and Ui 1\ Uj = 0 for all i j j. Let Pi denote the band projection of F onto TUidd, and write To for the operator Tuo' defined as in (1.4) above. Define R = Po' .. Pn-ITo, and S = Po' .. Pn- l (T - To). Then R is a lattice homomorphism and S is (n - I)-disjoint.

Proof. It is clear that Rand S are linear. Suppose x, y E E and x 1\ y = O. For all positive integers m, UI, ... , Un-I, X 1\ mUO, y 1\ muo are n + 1 mutually pairwise disjoint positive elements of E. Hence

TUI 1\ ... 1\ TUn-1 1\ T(x 1\ muo) 1\ T(y 1\ muo) = 0,

from which it follows, in succession, that

o = TUI 1\ ... 1\ TUn-I 1\ Tox 1\ Toy.

0= Pl ... Pn-I(Tox 1\ Toy)

= (Pl, .. Pn-ITOx) 1\ (Pl, .. Pn-ITOY).

0= Rx 1\ Ry.

R is a lattice homomorphism as claimed. Now suppose that Xl, ... ,Xn are n mutually pairwise disjoint positive ele

ments of E. It is sufficient to show that

n

/\ poeT - TO)Xi = O. i=l

(2)

Let m be a positive integer; write X = Xl + ... + Xn, and observe that Uo - Uo 1\

(l/m)x), and the n elements Xi -Xi 1\ muo, are n + 1 mutually pairwise disjoint

36 S.l BERNAU

positive elements of E. It follows that

n n

Tuo 1\ 1\ (T - TO)Xi ::::; Tuo /\ 1\ T(Xi - Xi /\ mUO) i=l i=l

n

::::; (T(uo - (l/m)uo 1\ X)) /\ 1\ T(Xi - Xi 1\ mUO) i=l

n

+ (l/m)T(uo /\ X) /\ 1\ T(Xi - Xi /\ mUO) i=l

n

= 0+ (l/m)T(uo /\ x) /\ 1\ T(Xi - Xi /\ mUO) i=l

::::; (l/m)TuO.

We conclude that

n

TuO /\ 1\ (T - TO)Xi = 0, i=l

and hence

n n

1\ poeT - TO)Xi = Po 1\ (T - TO)Xi = 0. i=l i=1

The proof is finished. Q.E.D.

THEOREM 1.6 Let E and F be vector lattices with F Dedekind complete, and T a positive n-disjoint linear operator from E to F, then there exist n lattice homomorphisms, T1 , . .. , Tn, from E to F, such that T = L:f=l Ii.

Proof The proof is by induction on n. The case n = 1 is a tautology. Suppose the theorem is true for n-l. By Zorn's Lemma there is a maximal set

P of band projections on F such that: if Pr, P2 E P and PI f P2 , then PI P2 = 0; and for each PEP there exist n mutually pairwise disjoint positive elements uo, ... ,Un-I E E such that P F c (Tuo /\ ... /\ TUn_l)dd. By Lemma 1.5, for each PEP there exists a lattice homomorphism TI (P) such that TI (P) =

PTI(P), and PT - T1(P) is (n - I)-disjoint. For 0 < X E E, define

Tlx = V{TI(P)x: PEP}.

If x, Y E E and X /\ Y ?: 0, we see easily that Tlx + Tly = TI (x + y). It follows that TI extends naturally to a linear operator from E to F. Suppose that x, y E E

VECTOR LATTICE HOMOMORPHISMS

and x /\ Y = 0, then

T1x /\T1y = V{T1(P)x: PEP} /\ V{T1(Q)y: Q E P}

= V {T1(P)x /\ Tl(Q)y: P, Q E P}

= V {T1(P)x /\ T1(P)y: PEP}

=V{T1(P)(x/\y): PEP}

=0.

Thus TI is a lattice homomorphism.

37

Let S = T - T1• We must show that S is (n - I )-disjoint. To this end we suppose that Uo, . .. ,Un-l E E and Ui /\ Uj = 0 for all i # j. By maximality of P we have

Tuo /\ ... /\ TUn-1 = V{P(Tuo /\ ... /\ TUn-I) : PEP}

and hence

Suo /\ ... /\ SUn-1 = V {P(SUO /\ ... /\ Sun-d : PEP}

= V{PSuo /\ ... /\ PSun-d : PEP}

=0

since each PS = PT - TI(P) is (n -I)-disjoint. Q.E.D.

COROLLARY 1.7 Let E and F be vector lattices with F Dedekind complete, and Tan n-disjoint linear operator from E to F. Then there exist n disjointness preserving linear operators T I , . .. ,Tn, from E to F, such that T = L:i=l 'no

Proof We check that ITI is n-disjoint and hence a sum of n lattice homomorphisms; say, ITI = SI + ... + Sn. Note that SI /\ T+ - SI /\ T-, is disjointness preserving. The proof is completed using the Riesz decomposition theorem. Q.E.D.

The method of proof above makes it clear that the decomposition of an n-disjoint linear operator as the sum of n lattice homomorphisms is highly nonunique. The next example shows this rather dramatically.

EXAJ\.1PLE 1.8 Let E = F = R2, and define T by T(a, b) = (a+b, a+b). Then T is 2-disjoint (by default since dim E = 2). Now let I be the identity map of E to F and define R from E to F by R(a, b) = (a, a). I and T - I are lattice homomorphisms, both of rank 2; also Rand T - R are lattice homomorphisms, both of rank 1. In fact this example can easily be modified to apply to any T E L(R2) of the form T(a, b) = (aa + {3b, ,a + 8b) with a, {3, " and 8 all positive. The relevant equalities are

a(a, ,) + b({3, 8) = (aa + {3b, ,a + 8b) = (aa,8b) + ({3b, ,a).

38 S.J. BERNAU

For related considerations and an example showing that some form of completeness is essential for our Theorem 1.6, even for n = 2, see [6].

2. Extensions of Vector Lattice Homomorphisms

A discussion of Hahn-Banach type extension theorems for vector lattice homomorphisms.

A survey of work by Z. Lipecki, D. Plachky, and W. Thompsen

W.A.l. Luxemburg and A.R. Schep the author

G.J.H.M. Buskes and A.c.M. Van Rooi}.

Here our basic set up is the following: E is a vector lattice, F is a Dedekind complete vector lattice, M is a vector sub lattice of E and T is a lattice homomorphism of Minto F. In addition we (almost always) assume that M is cofinal in E (for each x in E there exists m in M such that Ixl < m). A theorem of Hahn-Banach type would allow us to claim that there exists a lattice homomorphism S of E into F such that T is the restriction of S to M.

Completeness of F allows us to see that a dominating sublinear function p from E to F will allow a p-dominated extension of T. It is also easy to see that cofinality of M allows the construction of a sublinear dominant for T. Indeed this is essentially due to Kantorovich [9], and has been known almost as long as the Hahn-Banach theorem itself. The proof is identical to the classical HahnBanach proof. Unfortunately, all we can deduce from this is the existence of positive extensions of T. The theorem we would like is the following [10], [11], [12], [14], [15], [16].

THEOREM 2.1 Let E be a vector lattice, F a Dedekind complete vector lattice, M a co final vector sublattice of E, and T a lattice homomorphism of Minto F. Then T admits a lattice homomorphic extension to M.

By now there are many approaches to the proof of this theorem. A natural first choice of method is to try to mimic the classical proof of the Hahn-Banach theorem. An automatic maximal argument leads to a lattice homomorphic extension with maximal domain. All we then need is a constructive extension to the vector sublattice obtained by adjoining one element to the maximal domain. The way to do this is not immediately obvious, so the first proofs given for this theorem have used a variety of alternative methods, all interesting in their own right. We include summaries of several of the main ideas, and provide references which will enable the interested reader to find most of the others.


2.1. THE METHOD OF LUXEMBURG AND SCHEP

These authors deal with the difficulties of the one step extension of a vector lattice homomorphism from the sub lattice M to the sublattice generated by M and one additional element by discovering an entirely different technique. Their approach has considerable interest in its own right. It begins with a result which is an obvious corollary of Theorem 2.1. We provide here no more than a sketch of their methods; for details we refer to [16].

THEOREM 2.2 If M is Dedekind complete, M is a Riesz retract of E.

(I.e. there is a lattice homomorphic projection, p, of E onto M.)

COROLLARY 2.3 If M is Dedekind complete, Theorem 2.2 is true.

The extension is provided by Tp.

THEOREM 2.4 If T is order continuous, Theorem 2.1 is true. Proof The Dedekind completion, E, of E contains (an isomorphic copy

of) the Dedekind completion, M, of M. By order continuity, T extends to M. Corollary 2.3 now applies. Q.E.D.

To prove Theorem 2.1 we now proceed as follows. Choose an ideal J in E, maximal with respect to the property J n M = kerT. The lattice homomorphism induced by T on M / J is order continuous as a map into the Dedekind completion of T M. By Theorem 2.2 it lifts to a lattice homomorphism of E / J. Preceding this with the quotient map gives a lattice homomorphism of E lifting T into the Dedekind completion of T M. Finally we construct a lattice homomorphism of the Dedekind completion of T Minto F which leaves T M fixed.

2.2. THE METHODS OF LIPECKI, PLACHKY, AND THOMPSEN

There are many variations of a Hahn-Banach type proof of Theorem 2.1 due to Lipecki, either alone or with Plachky or Thompsen [10], [11], [12], [14], [15]. All depend, in one way or another, on determining conditions that force a positive extension of T to be an extreme point of the set of all positive extensions, and hence a lattice homomorphism. We begin with a theorem due to many authors. The result itself may be found in [3], [10], [15]. Our proof here is an outline of the method of [3].

THEOREM 2.5 The set of positive extensions ofT with domain E is convex and its extreme points are precisely the lattice homomorphic extensions.

Proof Suppose first that S is a positive extension of T which is not a lattice homomorphism. If u, v E M, x, Y E E and x A y = a = u A v, then

IS(u+x-v-y)l2:: SxASy.

40

If Sx /\ Sy ic 0, the linear map defined on span{ M, x - y} by

R(m + a(x - y)) = a(Sx /\ Sy)

S.J. BERNAU

is nonzero and admits an extension R to the whole of E such that Rz ::; Slzl. Then S ± R are positive extensions of T and S is not an extreme extension.

Conversely, suppose that S is a lattice homomorphic extension, P and Q are positive extensions, 0 < r < 1, and S = rP + (1- r)Q. Then P < Sir, so that P is a lattice homomorphism and for 0 ::; x ::; U, with x in E and U in M, we have

I(P - S)xl = /\ I(P - S)(x - ~u)1 n

::; (1 + I/r) /\ S (I(x - ~U)I) 1 + l/r S

::; 2n u.

Thus Px = Sx for all x in the ideal of E generated by M; so P = S = Q, and S is extreme. Q.E.D.

An individual result of Lipecki [11] can be obtained by strictly classical extension methods as follows. The first Lemma has a flavor of the proof of Theorem 2.5 above.

LEMMA 2.6 ([14]) A positive extension of T is extreme and hence a lattice homomorphism if and only if

/\{S(lx - ml) : mE M} = O.

THEOREM 2.7 A maximal extension S of T which has the property (2.6) is a lattice homomorphism of E into F.

We now consider a different approach which is more like the classical extension proofs. This material is contained in [13]. For all x in E define

p(x) = /\{Tm: mE M and x::; m}.

Then p is sublinear, agrees with T on M, and p(m + x) = Tm + p(x) for all m E M and x E E. Take x tj:. M, let W be the set of finite suprema of elements of the form m + rx, with m E M and r nonnegative real. Then W is a cone, closed under the formation of finite suprema, and p is additive and supremum preserving on W. Consider the set W - W. It is routine to show that this is the vector sublattice, Mo, generated by M and x, and that p extends, by p( WI - Wz) = PWI - pwz, to a linear transformation of Mo into F. Furthermore,

(p(WI-WZ))+ = (PWI-PWZ)+ =PWI Vpwz-pwz =P((WI-WZ)+)


so we have shown how to extend a lattice homomorphism to the vector sublattice generated by adjoining one element. Theorem 2.1 can now be proved by a classical Hahn-Banach argument.

2.3. THE METHOD OF THE AUTHOR

In the discussion above we constructed a sublinear dominant, p, for T. A classical Hahn-Banach argument provides a p-dominated linear extension to E. It is easy to see that the extension is positive, but impossible to guarantee that it is a lattice homomorphism. Another approach then is to produce a better dominant so that the dominated linear extension is forced to be a lattice homomorphism. For this it is only necessary to require the dominant to produce disjoint images for disjoint arguments. This is the approach followed in [3].

Write B(F) for the complete Boolean algebra of polar subspaces (disjoint complements) of F. Define a map A : M+ ---+ B(F) by

A(m) = (Tm)dd.

Then A is a lattice homomorphism of M+ into B(F).

THEOREM 2.8 A admits an extension, also denoted by A, to a lattice homomorphism of E+ into B(F).

This theorem extends an old theorem of Sikorski [20] and is proved by classical Hahn-Banach methods.

Before outlining its proof we note that W.A.J. Luxemburg has shown (private communication) that our Theorem 2.8 is logically equivalent to Sikorski's theorem. It is therefore stronger than the prime ideal theorem; and it is not known whether it is equivalent to the axiom of choice.

Proof of Theorem 2.8. For technical convenience least and great elements are adjoined to M+. A maximal extension is then obtained from a routine Zorn's Lemma argument. The existence of one step extensions is shown as follows.

If L is a sub lattice of E+ and x t/:. L the sublattice generated by L and x is {(x J\ u) Vv: u,v E L}. Let

t(x) = I\{A(m) V A(n)' : m ~ x J\ n}

and define

A((x J\ u) V v) = (t(x) J\ A(u)) V A(v).

The proof is finished. Q.E.D.

We now get a bonus. Once A is defined we can actually construct a lattice homomorphic extension of T with no additional maximal argument. We also get a pseudo-uniqueness theorem for the extension. It is completely determined by the polars of the positive elements. The details of the following outline may be found in [3].

42 S.l BERNAU

DEFINITION 2.9 For x E E, a bounded decomposition of x is a finite sequence ((Xl, ml), ... , (Xn, m n)) such that X = Xl + ... + Xn, and for each i, Xi E E, mi EM, and ° ~ Xi ~ mi.

DEFINITION 2.10 For each X E E+ write Px for the band projection of F onto A(x) and define for X E E+

where the infimum is taken over all bounded decompositions of x.

THEOREM 2.11 The map S defined above is additive and agrees with T on M+. Its extension, by differences, to E is a lattice homomorphism extending T.

Proof Subadditivity and positive homogeneity of S are easy. If X 1\ Y = 0, A(x) 1\ A(y) = 0, so Sex) 1\ S(y) = 0, and the lattice homomorphism property follows.

Suppose m E M+ and ((Xl, mI), ... , (Xn, m n )) is a bounded decomposition of x. We prove that Z=~l PxiTmi 2: Tm. The case n = I is clear. If n = 2 we have

m - m 1\ mi ~ Xj (i # j)

from which we conclude

and hence,

Tm = PX1 (1 - PxJTm + PX2 (1 - PX1 )Tm + PX1 PX2 Tm

~ PX1 (1 - Px2 )Tmj + Px2 (1 - Px1)Tm2 + PX1 PX2 T(mj + m2)

= Px1Tml + Px2Tm2.

For larger n the argument is similar, but technically more complicated. Finally to see that S is additive on E+ we take first x, y E E+ and m E M

such that X + Y ~ m. Special choices of bounded decompositions of X and y enable us to show that

SeX) + S(y) ~ Px+yT(m)

and this result together with an application of the Riesz Decomposition Theorem finishes the proof. Q.E.D.

Our final result is the pseudo uniqueness theorem we mentioned above.


THEOREM 2.12 If U and V are lattice homomorphic extensions of T and (Ux)dd = (Vx)dd for all x E E+, then U = V.

The proof depends on the observation that if we define A as A (x) = U xdd for all x EM, and then construct S as above, we have U ::; S ; since M is cofinal in E this forces U = S and since V agrees with U on M, we similarly have V=S.

2.4. THE METHOD OF BUSKES AND VAN ROOIJ

These authors have probably the most transparent existence proof of all. Their paper [5] is written from a slightly more general perspective than we need here, but we give the essential features. Their insight was to find a way exploit the fact that the natural sublinear dominant, p, used in the two preceding sections, is a V -homomorphism (Le., p( x V y) = p( x) V p(y) and also agrees with T on M. From that point on there is no further need to have M cofinal in E. For the rest of this discussion we drop the assumption that M is cofinal in E, and assume that M is a vector sublattice.

LEMMA 2.13 The set S of sublinear V -homomorphisms rp on E into F such that rplM = T, contains minimal elements.

It is routine to verify that the pointwise infimum of a downwards ordered chain in S is an element of S.

THEOREM 2.14 Each minimal element of S is an extension of T to a lattice homomorphism of E.

Proof Let rp be a minimal element of S. Fix Y E E and define

A(X) = inf{rp(x + ay) - arp(y) : a 2': O} (x E E).

For 0 ::; a ::; (3 we have (3rp(y) = rp({3y) ::; rp(x + (3y) + rp( -x), whence

-rp( -x) ::; rp(x + (3y) - (3rp(y)

::; rp( x + ay) + ({3 - a )rp(y) - (3r.p(y)

= rp(x + ay) - arp(y).

Thus, A(X) 2': -rp( -x) for all x, and we can check easily that A E S. By minimality A = rp. We can now conclude that rp(x + y) - rp(y)' = rp(x) for all x, y E E. Theorem 2.14 follows. Q.E.D.

Buskes and van Rooij have another extension theorem which we also mention.

THEOREM 2.15 Let M be a vector sublattice of E and F a Dedekind complete vector lattice. Suppose T is a vector lattice homomorphism of Minto F and that

44 S.J. BERNAU

i.p is a positive sublinear V-homomorphism of E into F such that Tx :::; i.p(x) for all x E M. Then T admits an extension to a lattice homomorphism S of E into F such that Sx :::; i.p(x) for all x E E.

The proof of this result is a technical modification of the classical HahnBanach dominated proof which takes care to see that that the constructed dominated extension is an extreme point of the set of i.p-dominated extensions. The lattice homomorphism property is a consequence of the extreme point property. The details become rather complicated and it would be desirable to construct a proof which required less by way of representation theory. Perhaps, another ingenious algebraic minimality maximality argument can be found to achieve this.

Acknowledgment

This paper is based on the lecture delivered by the author at the third annual meeting of the Caribbean Mathematical Foundation in June 1990. The author thanks the conference organizers most sincerely for their generous invitation and support to attend this meeting.

References

1. C.D. Aliprantis, o. Burkinshaw, Positive Operators, Acad. Press, Orlando, 1985. 2. s. 1. Bemau, Orthomorphisms of Archimedean vector lattices, Math. Pmc. Camb. Phil.

Soc. 89 (1981), pp. 119-128. 3. S. 1. Bemau, Extensions of vector lattice homomorphisms, 1. London Math. Soc. (2) 33

(1986), pp.516-524. 4. SJ. Bemau, C. B. Huijsmans, B. de Pagter, Sums of vector lattice homomorphisms,

Proc. Amer. Math. Soc. 115 (1992),151-156. 5. GJ.H M. Buskes, A.C.M. van Rooij, Hahn-Banach for Riesz homomorphisms, Nederl.

Akad. Wetensch. Pmc. Ser. A 92 (1989), pp. 25-34. 6. David C. Carothers, William A. Feldman, Sums of homomorphisms on Banach lattices,

1. Operator Theory 24 (1990),337-349. 7. Michel Duhoux, Mathieu Meyer, A new proof of the lattice structure of orthomorphisms,

1. London Math. Soc. (2) 25 (1981), pp. 375-378. 8. C. B. Huijsmans, B. de Pagter, , Disjointness preserving and diffuse operators, Compositio

Math. 79 (1991), pp.351-374. 9. L. V. Kantorovitch, Concerning the problem of moments for a finite interval, Dokl. Acad.

Nauk SSSR 14 (1937), pp. 279-284. 10. Z. Lipecki, Extensions of positive operators and extreme points. II, Colloq. Math. 42

(1979), pp. 285-289. 11. Z. Lipecki, Extensions of vector lattice homomorphisms, Proc. Amer. Math. Soc. 79

(1980), pp. 247-248. 12. Z. Lipecki, Extensions of positive operators and extreme points. III, Colloq. Math. 46

(1982), pp. 263-268. 13. Z. Lipecki, Extension of vector lattice homomorphisms revisited, Nederl. Akad. Wetensch.

Pmc. Ser. A 88 (1985), pp. 229-233. 14. Z. Lipecki, D. Plachky, W. Thompsen, Extensions of positive operators and extreme

points. I, Colloq. Math. 42 (1979), pp. 279-284.


15. Z. Lipecki, W. Thompsen, Extensions of positive operators and extreme points. IV, Colloq. Math. 46 (1982), pp. 269-273.

16. W.AJ. Luxemburg, A.R. Schep, An extension theorem for Riesz homomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 82 (1979), pp. 145-154.

17. W.A.J. Luxemburg, A.C. Zaanen, Riesz Spaces, I, North-Holland, Amsterdam, 1971. 18. M. Meyer, Le stabilisateur d'un espace vectoriel n~ticule, C. R. Acad Sci. Paris Sir. A

283 (1976), pp. 249-250. 19. H.H. Schaefer, Banach Lattices and Positive Operators, Grundlehren der Mathematischen

Wissenschaften 215 , Springer, Berlin, 1974. 20. Roman Sikorski, A theorem on extension of homomorphisms, Ann. Soc. Pol. Math 21

(1948), pp. 332-335. 21. A.C. Zaanen, Riesz Spaces, II, North-Holland, Amsterdam, 1983.


Baillon's Theorem on Maximal Regularity

B. EBERHARDT and G. GREINER Mathematisches lnstitut der Universitiit Tiibingen, Auf der Morgenstelle 10, 7400 Tiibingen, Germany

(Received: 27 Apri11992)

47

Abstract. The aim of this note is to give a proof of Baillon s Theorem on Maximal Regularity. Though it is in some sense a negative result (it states that for abstract Cauchy problems maximal regularity can occur only in very special cases), it is commonly accepted that it is important. Many people believe that its proof is very complicated. This might be due to the fact that Baillon's note in the Comptes Rendus is rather short and sometimes difficult to understand. The proof outlined here follows basically Baillon's lines. However it is simplified and (hopefully) easier to understand.

Mathematics Subject Classifications (1991): 47D03, 34A12

Key words: Baillon's theorem, abstract Cauchy problem, strongly continuous semigroup

A strongly continuous semigroup T = (T(t)k~.o on a Banach space X is said to have the maximal regularity property (MR) if the following condition is satisfied:

For every f E C([O, T], X) the convolution T * f which is defined by (T * J)(t) := fJT(t - s)f(s) ds is continuously differentiable.

It is not difficult to verify that whenever (MR) is true for some T > 0, then it is true for every T. Thus we can assume without loss of generality that T = 1. From the identity

1 h((T * J)(t + h) - (T * J)(t))

1 1 (h = h(T(h) - Id)(T * J)(t) + h io T(s)f(t + h - s) ds

it follows that for a continuous f we have T * f E Cl([O, 1], X) if and only if T * f E C([O, 1], Xl) where Xl is the Banach space D(A) equipped with the graph norm. 1 Thus (MR) can be restated as follows:

1 Note that a function which is differentiable from the right and has a continuous right side derivate is actually c 1•

48 B. EBERHARDT AND G. GREINER

For every f E C([O, TJ, X) the convolution T * f is a continuous function from [0, T] to Xl.

The convolution T * f is of interest, since it is the only possible solution of the inhomogeneous Cauchy problem (CP)

u(t) = Au(t) + f(t), u(O) = 0.

More precisely, whenever u E CI([O, 1], X) nC([O, 1], D(A)) satisfies (CP) then u=T*f.

Trivial examples of semi groups satisfying (MR) are those with a bounded generator. Then t f.-...* T(t) is Coo on the interval [0, T] which implies that T * f is Coo for every f E C([O, TJ, X). An example of an unbounded generator A satisfying (MR) is the multiplication operator on Co, the space of all null sequences, defined by A(en) := (-n· en). The corresponding semigroup is given by T(t)x = (e-ntxn)nEN for x = (xn)nEN E Co· Given f E C([O, 1], co) then f = Un) with fn E C[O, 1] and limn--->cx:> Ilfnll = 0. A straightforward calculation shows that T * f =: 9 = (gn), where gn E C[O, 1] is given by gn(t) := J~ e-n(t-s) fn(s) ds. It follows easily that get) E D(A) for all t ;::: ° and that g: [0, 1] -+ D(A) is continuous at every t > 0. In order to show continuity at t = ° we apply the (second) mean value theorem on integrals and obtain

-ngn(t) = -n lot e-n(t-s) ds· f(en,t) = -(1 - e-nt ) . f(en,t)

for suitable ~n,t E [0, t]. Then given E > 0, I - ngn(t) I s 1 . Ilfnll < E for n ;::: N = N(E) (uniformly in t E [0,1]). Moreover, for n < N = N(E) we have I - ngn(t) I s (1 - e-Nt)llfnll < E for t sufficiently small. We conclude that IIAg(t)ll-+ ° as t -+ 0, hence g: [0, 1] -+ D(A) is continuous at ° as well.

In the example mentioned above the choice of the space Co was crucial! As a consequence of Baillon's Theorem this cannot be true in V-spaces. In fact, the result states that unbounded generators which have (MR) can only exist in Banach spaces containing a closed subspace which is isomorphic to Co. We need the following characterization of Banach spaces containing Co.

THEOREM 0.1 A Banach space X contains a closed subspace which is isomorphic to Co if and only if there exist a sequence (xn) C X and a constant M such that

inf {llxnll} > ° nEN

and Ilxo ±XI ± X2 ± ... ±xnil S M

for every n E N and all possible choices of signs + or -. (1)

The proof follows from joint work of C. Bessaga and A. Pelczyfiski [2, Coroll. 1 and Lemma 3]. We sketch a direct proof in the appendix.

Another ingredient for the proof is the following result of Hille (cf. [5] or [6, 2.5.3]).

BAILLON'S THEOREM 49

THEOREM 0.2 Let (T(t) be a Co-semigroup on X with generator A. If for every x E X the mapping t 1--+ T(t)x is differentiable on (0,00) and lim SUPt-->o t IIAT(t)11 < ~ then A is a bounded operator.

COROLLARY 0.3 If (T( t)) is a semigroup with an unbounded generator A satisfying (MR) then ImT(t) C D(A) for t > ° and limsuPt-->o tIIAT(t)11 2:: ~.

Proof For x E X we consider the function f(t) := T(t)x. Then (T* f)(t) = t . T(t)x. By (MR) this function is C l , hence t 1--+ T(t)x is CIon (0,00). It follows that ImT(t) C D(A) for t > 0. Moreover, the theorem implies limsuPt-->o tIIAT(t)11 2::~. Q.E.D.

A function f: [0, 1] --+ X is said to be piecewise continuous if f is continuous except at finitely many points ° < tl < '" < tn < 1 and such that rightand lefthand limits exist at every point ti. The set of all piecewise continuous functions will be denoted by Cpw([O, 1], X).

We will show that for a semigroup T satisfying (MR) and a piecewise continuous f the convolution T * f is a continuous mapping into Xl := (D(A), II·IIA).

PROPOSITION 0.4 If T satisfies (MR) and f E Cpw([O, 1], X), then T * f E

C([O, 1], Xd. Moreover, there is a constant C such that

sup IIA(T* f)(t)11 :S C· sup Ilf(t)11 for all f E Cpw([O, 1], X)(2). 09:::;1 09:::;1

Proof We only consider the case where f has one discontinuity at tl say. The functions t 1--+ f(t) and t 1--+ f(tl + t) defined on [0, td and (0,1 - td have continuous extensions fo and !I say. Then

{ (T * fo)(t) if t E [0, til, (T*f)(t) = (T(t - tI)(T * fO)(tl)) + (T * fI)(t - tl) if t E (tl' 1]. (3)

It follows that both T * f and A(T * f) are continuous. Thus T * f E C ([0, 1], Xl)' First we observe that the mapping f 1--+ T * f is continuous from

(Cpw([O, l],X), 11·1100) into (C([O, l],X), 11·1100)' In fact this follows from the estimate

II(T * f)(t) II :S lot IIT(t - s)11 Ilf(s)11 ds :S (lol IIT(s)11 dS) Ilflloo.

The considerations above show that the range of this mapping is contained in the Banach space C([O, 1], Xt) which is continuously embedded in C([O, 1], X). Thus by the closed graph theorem f 1--+ T * f is continuous from Cpw([O, 1], X) into C([O, 1], Xl)' It follows that there is a constant C such that (1) holds. Q.E.D.

Now we have all the prerequisites in order to prove the main result.

THEOREM 0.5 (Baillon's Theorem) Let A be the generator of a Co-semigroup (T(t)) on a Banach space X satisfying (MR). Then either A is bounded or X contains a closed subspace which is isomorphic to Co.


Proof We assume that (A, D(A)) is an unbounded generator of a Co-semigroup (T(t)) on a Banach space X satisfying (MR). In order to prove the Theorem we have to show that X contains a sequence (xn) with the properties stated in Theorem 0.1. Q.E.D.

Construction of the Sequence. Because of the Corollary to Theorem 0.2 we can find a sequence of positive real numbers (ti)iEN such that

to:= 1, 1

ti < 2i ti-I for every i = 1,2,3, ... , (4)

and

for every i = 1,2,3, .... (5)

Then there are elements Yi EX, IIYillx :s: 1 such that IltiAT(ti)Yillx > de for i = 1,2,3, ... and AT(I)yo f 0. From Proposition 0.4 we deduce that for all i

IltiAT(ti)Yillx = IIA(T * Yi)(ti) IIx :s: C ·llYill :s: C, where Yi(t) := T(t)Yi, C := C . sUPO<t<1 IIT(t) II.

If we define: - -

we have at once

inf Ilxi II > 0. i2':O

for i = 0, 1, 2, ... ,

(6)

(7)

(8)

It remains to verify the second condition of (1). We therefore choose n E Nand Ei with Ei = ± 1 for i = 0, 1, ... n. Defining the following piecewise continuous function:

we obtain

1 - ti :s: s :s: 1 - ti+ 1 ,

i = 0, ... ,n -1, 1 - tn :s: s :s: 1,

(T * 1)(1) = 101 T(1 - s)f(s) ds

n-I rl - t i+ 1

= ~ Ei JI-ti T(1 - s)T(s + ti - I)Yi ds

+ En rl T(1 - s)T(s + tn - I)Yn ds JI-tn

n-I = L Ei(ti - ti+I)T(ti)Yi + EntnT(tn)Yn.

i=O

BAILLON'S THEOREM

Therefore we get using (6), (7) and the triangle inequality:

IIEoxo + EIXj + ... + EnXn - A(T * f)(l)llx :::; IIEotIAT(l)yo + Elt2AT(tdYl

+ ... + En-ltnAT(tn-l)Yn-l + Ollx

= IIEOt1XO + El t2 Xl + ... + En-l ~Xn-lll tl tn-l x

1- 1- 1 - -:::; 2C + 4C + ... + 2n C :::; C.

Together with the inequality of Proposition 0.4,

IIA(T * f)(1)11 :::; Cllfll :::; C· sup IIT(t)11 099

we obtain

IIEoxo + EIXI + ... + Enxnllx :::; IIEoxo + EIXI + ... + EnXn

- A(T * f)(l)llx + IIA(T * f)(1)11 :::; C + C· sup IIT(t)11 < 00,

099

independently of n EN and the choice of Ei = ±1. Q.E.D.

51

A few words to the consequences of Baillon's Theorem. Recall that closed subspaces of reflexive Banach spaces are also reflexive. Therefore on reflexive Banach spaces all semigroups satisfying (MR) are uniformly continuous (and therefore not very interesting). More generally, from the fact that Co is not weakly sequentially complete (the sequence (L::i=l ei)nEN (ei the i-th unit vector) is weak Cauchy but not weakly convergent), it follows that on weakly sequentially complete spaces (MR) can occur only when the generator is bounded. In addition to reflexive spaces the Ll-spaces are weakly sequentially complete ([4, IV.8.6]). The Sobolev spaces Wp,k can be considered as closed subspaces of products of LP -spaces. Hence they are weakly sequentially complete as well.

For example the semigroup generated by the multiplication operator A(en) =

(-n· en) is maximal regular on Co but not on fP, 1 :::; p < 00.

Or the other way around, if you have an unbounded generator on an LP -space (1 :::; p < (0), a Sobolev space or a reflexive Banach space (e.g. Hilbert space), then there will be always a continuous function f E C([O, TJ, X) such that the inhomogeneous Cauchy problem

du dt = Au+ f,

u(O) = 0

has no classical solution.


Appendix

In this Appendix we want to give a direct proof of Theorem 0.1. It is based on the following lemma on infinite matrices A = (aij)i,jEN' A submatrix of A is an infinite matrix B = (bij)i,jEN obtained from A by cancelling some rows and the corresponding columns. In other words, there is a subsequence (nj) j EN of the natural numbers such that bij = aninj . In the following we consider matrices which give rise to bounded linear operators on £1. Thus

all columns are elements of £1 and the £I-norm of the columns is uniformly bounded (*)

The norm of the induced operator is the supremum of the .el-norm of the columns of A.

LEMMA 0.6 Let A be an infinite matrix satisfying (*). If the diagonal (aii) does not converge to zero, then there is a submatrix B of A which induces an isomorphism on £1.

Proof. By assumption there is a subsequence (ni) and 6 > 0 such that infiEN{laninil} ~ 6 > O. Thus considering the submatrix defined by (ni) we can assume w.l.o.g. that infiEN{laiil} 2: 6 > O.

Now we show that for every E > 0 there is a sub matrix B of A which satisfies Lih Ibijl < E for every j EN. It follows that liB - DII ::; E where D denotes the diagonal part of B. In case E < 6 the matrix B is invertible, because its diagonal part is invertible with liD-III::; 6-1 and liB - DII ::; E < 6.

We construct B in two steps. 1) There is a submatrix C such that Li>j !cijl < ~ for every j. 2) There is a submatrix B of C such that Li<j Ibijl < ~ for every j.

Step 1)

Define nl := 1. Since the first column is £1 there is a subsequence (nl, n12, n13, ... ) of N such that L~2lanlill < ~. Define n2 := n12. Since the n~ column is .e1 there is a subsequence (nl,n2,n23,n24, ... ) of (nl,n12,n13, ... ) such that L~31an2in21 < ~. Define n3 := n23. Proceeding this way one obtains (recursively) a subsequence such that the corresponding submatrix C satisfies Li>j ICijl < ~ for every j.

Step 2)

Let c be a bound for the £1-norm of the columns. If we choose mEN such that m . ~ > c, then among the first m rows of B there must be one which contains infinitely many elements of absolute value less that ~. (Otherwise there are columns which have £I-norm greater than m· ~ > c which is a contradiction).

BAILLON'S THEOREM 53

Let nl be such a row and choose a subsequence(nl,n12,n13, ... ) of(nl,m +1, m + 2, , ... ) such that Ibnj,nlj I < % for j ~ 2.

For the same reason as above there is among the 2m rows n12, n13, ... , nl,2m+l one which has infinitely many elements of absolute value less than ~. Let n2 be such a row and choose a subsequence (n 1, n2, n23, n24, ... ) of (n 1 , n2, n 1 ,2m+2, ... ) such that I bn2 ,n2j I < ~ for all j ~ 3. Proceeding this way one obtains (recursively) a subsequence such that the corresponding submatrix B of C satisfies Ibijl < 2-i- 1E for every j > i. Hence L.i<j Ibijl < L.{;;:f 2-i- 1E < ~ for every j. Q.E.D.

Now we can give the Proof of Theorem 0.1. Let (Xn) be sequence in the Banach space X such

that 6 := infnEN Ilxnll > 0 and Ilxo ±X1 ±X2 ± ... ±xnll :S M for every n E N and all possible choices of signs. By the Hahn-Banach theorem there exist linear functionals x~ E X' such that

Ilx~11 = 1 and for all n E N.

The infinite matrix A = « Xi, xj > )ijEN satisfies the hypotheses of the

Lemma. In fact, lajjl = Ilxjll ~ 6 and L.~o laijl = L.~o I < Xi,Xj > I (L.~o EiXi, xj) where Ei = sgn( < Xi, xj ». It follows that L.i=o laij I :S II L.i=o EiXi II . Ilxj II :S M for every n, j E N.

According to the Lemma we choose a subsequence (ni) which defines an invertible submatrix B of A. Let Yi := xni ' Y~ := X~i'

We define a linear mapping To from the space of all finite sequences rp into X by TO(~n) := L.n ~nYn. We claim that To is bounded. In fact, each L.n ~nYn with (~n) E rp, 11(~n)11 :S 1 is a convex combination of vectors ±XO±XI ±X2±'" X m. Since each of these vectors has norm less than M so has L.n ~nYn. It follows that To is bounded and has norm :S M. To can be uniquely extended to a bounded linear map T: Co -+ X.

Furthermore we define 5: £1 -+ X' by 5(Tfn) := L.n TfnY~. Obviously 5 is linear and bounded (11511 :S 1) and it is easily verified that the composition T' 0 5: £1 -+ X' -+ £1 is represented by the matrix B. Thus T' 05 is invertible and therefore its adjoint 5' 0 T" as well. For ~ E Co C goo we have IIT~II =

IIT"~II ~ 115'11-1115' 0 T"~II ~ 115'11-111(5' oT")-111-111~11. This shows that Tis an isomorphism of Co onto a subspace of X. Q.E.D.

References

1. J.B. Baillon, Caractere borne de certains generateurs de semigroupes lineaires dans les espaces de Banach, C.R. Acad. Sc. Paris 290 (1980), pp. 757-760.

2. C. Bessaga, A. Pelczyriski, On bases and unconditional convergence of series in Banach spaces, Studia Math. XVII (1958), pp. 329-396.

3. Ph. Clement (et a!.), One-Parameter Semigroups, CWI Monographs 5, North-Holland, Amsterdam, 1987.

4. N. Dunford, J.T. Schwartz, Linear Operators, /, Wiley, New York, 1958.


5. E. Hille, On the differentiability of semigroups of operators, Acta Sc. Math. (Szeged) 12 (1950), pp. 19-24.

6. A. Pazy, Semigroups of Linear Operators and Applications to Panial Differential Equations, Springer-Verlag, New York, 1983.

Acta Applicandae Mathematicae 27: 55-65, 1992. @ 1992 Kluwer Academic Publishers.

Fraction-Dense Algebras and Spaces

A.W. HAGER Department of Mathematics, Wesleyan University, Middletown, CT 06457, U.S.A.

and

JORGE MARTINEZ Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A.


55

Abstract. A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fraction-dense f -rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For Archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fraction-dense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover.

Mathematics Subject Classifications (1991): 13B30, 54C99

Key words: commutative ring, fraction-dense ring, fraction dense space

Introduction

The motivation for the concepts of fraction-dense algebras and topological spaces introduced here comes from examining the relationship between the classical and maximal ring of quotients of a commutative ring with identity.

Unless further qualified, every ring in this exposition will be commutative, possess an identity, and also be semi-prime, in the sense that there are nonzero nilpotent elements.

An I -ring is a lattice-ordered ring in which a 1\ b = 0 implies that a 1\ be = 0 for each e 2: O.

Likewise, all topological spaces are assumed to be Tychonoff, unless the contrary is expressly stated.

Recall that a Hausdorff space is Tychonoff if the cozero-sets (of real-valued continuous functions) form a base for the topology. All lattice-ordered groups in this article are Abelian. Our standard references for this theory are [2] and [5].

Suppose that A is an I-ring; qA stands for its classical ring of quotients and Q A for its maximal ring of quotients. For the fundamental properties of these quotient rings we refer the reader to [21], [4] and [23]; the reader should

56 A.W. HAGER AND JORGE MARTINEZ

also consult [24] and [30]. The term 'ring of quotients' should be interpreted as follows: assume that A is a subring of the ring B; we say that B is a ring of quotients of A if for each pair b1, b2, with b2 =J 0, there is an a E A such that ab1 and ab2 both belong to A and ab2 =J 0. Each ring has a (unique) maximal ring of quotients; [21] treats the subject in full generality, whereas [4] gives a representational construction of the maximal ring of quotients of a semiprime ring. That is the procedure followed in [23] for f-rings, and there it was shown that QA can be given a lattice-ordering so that it becomes an f-ring, and contains A as an f-subring. qA also has a natural ordering making it an f-ring, and A ~ qA ~ QA. QA is the A-injective hull of A, and it is a von Neumann ring, in the sense that for each a E A there is an x E A such that a2x = a.

In considering the fractions in q A, first observe that in each fraction a I bone can assume without loss of generality that b > 0; then alb ~ 0 precisely when a ~ 0. From this we conclude that A is rigid in qA. (Recall that if H is a lattice-ordered group in which G is an f-subgroup, G is said to be rigid in H if for each h E H there is agE G such that hl..l.. = gl..l..; the symbol 1- stands for 'polar', and in any situation, such as this one of inclusion, is understood in the only possible way, namely as denoting polars in the larger object).

It is shown in [9] that if G is rigid in H, then the contraction map P --+ PnG is a homeomorphism from Min (H), the space of minimal prime £ -ideals of H, as a topological space with the hull-kernel topology, onto Min(G). Recall the wellknown fact that for semi-prime f -rings, 'minimal prime i-ideal' and 'minimal prime (ring) ideal' mean the same thing.

For any Tychonoff space X, recall from [11] that Q(X) == Q(C(X)) is the algebra of all continuous real-valued functions defined on some dense open subset of X, where it is understood that two such functions which agree on the intersection of their domains are identical.

By contrast, q(X) = q(C(X)) is the algebra of continuous real-valued functions defined on cozero-sets of X, with the same proviso for identification on common domains of definition.

The lattice-ordered group G is said to be orthocomplete if it is laterally complete-that is, every subset of pairwise disjoint elements has a supremumand also projectable. (A lattice-ordered group G is projectable if for each g E G, G = gl.. + gl..l...) The orthocompletion of a lattice-ordered group G, denoted by oG, is a lattice-ordered group which is orthocomplete, containing G densely, and such that no proper i-subgroup of oC contains G and is orthocomplete).

It is shown in [23, Thm. 1.5] that for any semi-prime f-ring A, QA contains the orthocompletion oA; moreover, QA = q(oA). If A is projectable or Archimedean, then the order of the operators q and 0 can be reversed; see [23, 1.4 and 1.8.1]. Anderson-Conrad show in [1] that for A = C(X),oA = QA. The proofs depend heavily upon the construction by Banaschewski in [4], and also on the analogous one for the orthocompletion in [6].

FRACTION-DENSE ALGEBRAS AND SPACES 57

In an off-hand manner it is also asserted in [1] that for A = C(X), qA = QA. Now this is not true, as one sees by taking the space ,8N \ N; it has no proper, dense cozero-sets (see [14, Chapt. 6]), and so q(,8N \ N) = C(,8N \ N), whereas the maximal ring of quotients is much bigger; since ,8N \ N is not extremally disconnected, it has plenty of dense open subsets U which are not C* -embedded; any function which belongs to C*(U) and cannot be extended continuously to ,8N \ N is in Q(,8N \ N).

And so the springboard for this article is the question: when is qA = QA? For reasons which we shall not motivate at this point, it is more interesting to ask the question: when is qA rigid in QA? The example we just gave is one, as we shall presently demonstrate, in which qA is not rigid in QA.

We proceed to examine J-rings A in which QA contains qA rigidly.

1. Fraction-Dense Spaces

We say that the semi-prime J-ring A isfraction-dense if qA is rigid in QA. If X is a Tychonoff space and A = C (X) then we say that X is a fraction-dense space if A is fraction-dense. If qA = QA, A will be called strongly fraction-dense; likewise X is a strongly fraction-dense space if C(X) is strongly fraction-dense. The reader should notice at the outset that the class of fraction-dense spaces is quite extensive; since in a metric space every open set is a cozero-set, it follows that Q(X) = q(X) for every metric space X. Indeed, every metric space is strongly fraction-dense.

Observe also that if X is an extremally disconnected space, then every (dense) open subset is C* -embedded (see [14, IH.6]), which implies that q(X) = Q(X) = D(X), the J-algebra of all continuous functions J defined on X with values in the extended reals, and for which fin(J) = {x E X: IIJ(x)11 < oo} is a dense subset of X. Thus, every extremally disconnected space is strongly fraction -dense.

Theorem 1.1, which we shall state presently, gives a number of criteria for A to be a fraction-dense J-ring. Before proceeding to it however, let us recall some definitions. We have already recalled that of projectable lattice-ordered groups; we say that G is strongly projectable if G = K..L + K..L..L, for each polar subgroup K. (Incidentally, we shall denote by P(G) the Boolean algebra of all polars of G, and by Pr(G) the sublattice generated by the principal polars g..L..L.)

We recall the definition in [8] of a complemented lattice-ordered group: G is said to be complemented if for each g E G there is an h E G such that IlgiiA Ilhll = 0 and Ilgll V Ilhll is a weak order unit. It is proved in [8] that G is complemented if and only if Min( G) is compact. (Compare this with the result for rings in terms of the Boolean algebra of annihilators in [17].)

It is not hard to see that A is complemented precisely when qA is projectable; this happens, in turn, if and only if qA is von Neumann. Observe as well, that since Q A is orthocomplete, it is also strongly projectable.


Before stating the theorem observe, finally, that G is a complemented latticeordered group if and only if Pre G), the lattice of principal polars of G, is a subalgebra of P( G). Moreover, in this event Pre G) is the Stone dual of Min( G). Thus, Pre G) is complete precisely when Min( G) is extremely disconnected, by the Stone-Nakano Theorem; (see [29, p. 47]). (Recall that a space X is extremely disconnected if the closure of each open set is open.) Appealing to [8, Thm. 2.7], we see that Min( G) is compact and extremally disconnected if and only if every polar of G is principal.

If A is a f -ring, then A( 1) stands for the convex f -subring generated by 1; we shall refer to it as the bounded subring of A. Note that A( 1) is rigid in A.

THEOREM 1.1 Suppose that A is a semi-prime f-ring. Then the following are equivalent:

1) A is fraction-dense. 2) A(l) is fraction-dense. 3) A is rigid in QA. 4) qA is strongly projectable. 5) Min(A) is compact and extremally disconnected. 6) A is complemented and every polar of A is principal. 7) qA and QA have the same idempotents.

Proof For any lattice-ordered groups G, Hand K,such that G is an €subgroup of Hand H is an €-subgroup of K, G is rigid in K if and only if G is rigid in Hand H is rigid in K. (See [8].) With this in mind it is immediate that 1) and 3) are equivalent.

If qA is rigid in QA, then since QA is strongly projectable, P(QA) =

Pr(QA), and therefore the same is true for qA. From this it is not hard to see that 3) implies 4).

Assuming 4), we have from earlier remarks that A is complemented, and from this that Min(A) is compact. As in the previous section, P(qA) = Pr(qA), and the same holds for A. Hence Min(A) is extremally disconnected. Thus 4) implies 5). From what has already been said, it is clear that 5) and 6) are equivalent.

Next we show that 5) implies 1). This follows from the rigidity of A in qA, along the lines of previous arguments: Min(qA) is compact and extremally disconnected, whence qA is strongly projectable. (So we have actually shown that 4) follows from 5).) Since QA is the orthocompletion of qA, this means that the contraction map P -+ P n qA is a Boolean isomorphism from P( QA) onto P(qA), all of which implies that each principal polar of QA contracts to a principal polar of qA. This shows that qA is rigid in QA.

As we now know that 1) is equivalent to statements 3) through 6), observe that they are all equivalent to 7) because, in a strongly projectable semi-prime f -ring, every polar is the principal polar of an idempotent.

Finally, note that A(l) is rigid in A, so that their spaces of minimal prime ideals are homeomorphic. This implies that 2) is equivalent to the rest, and the


proof of the theorem is complete. Q.E.D.

From Theorem 1.1 we have, right away, the following corollary; as usual, {3X denotes the Stone-Cech compactification of the space X.

COROLLARY 1.2 A Tychonoff space X is fraction-dense if and only if (3X is fraction-dense.

Observe that, since q(X) = q(C*(X)) and Q(X) = Q(C*(X)), we also get that X is stongly fraction-dense if and only if {3X is.

For the following corollary, let us first recall the notion of absolute of a space, as well as the concept of an irreducible map. It will be sufficient for our purposes to present the situation for compact spaces.

If f: X --+ Y is a continuous surjection (of compact spaces), then it is said to be irreducible if Y cannot be obtained as the image under f of a proper closed subset of X. Here are some basic observations about irreducible maps; the proofs may be found in [15] or [3]. To begin, note that the continuous surjection f: X --+ Y is irreducible if and only if the functorially induced embedding C(f): C(Y) --+ C(X), C(f)(g) = g. f, is an (order-) dense embedding. Also, recall that if f is irreducible, then the inverse image of a dense subset of Y is dense in X ([15, 2.7(a)]).

Now, for a given compact space, let R(X) denote the Boolean algebra of regular closed sets. This is a complete algebra: its Stone dual, EX, is therefore an extremally disconnected space. Viewing EX as the space of ultrafilters on R(X), we have a natural map ex: EX --+ X which assigns to the ultrafilter a the unique point of X common to all its members. (For further details see [29] or [26]. EX is sometimes called the Gleason space of X; it was first studied by A. Gleason in [12]. In this study the extremally disconnected spaces are viewed as projective spaces, and EX as the 'projective cover' of X.) We shall refer to EX as the absolute (space) of X.

Finally, note that the stipulations on f: E --+ X that f be irreducible and E be extremally disconnected, characterize the absolute of X, in the following sense: If f and E have the stated properties, then there is a homeomorphism g: EX --+ E such that fg = ex.

Now to the second corollary of Theorem 1.1. Let J (A) denote the Jacobson radical of A. It will also be most convenient to suppose that A satisfies the bounded inversion property: a > 1 implies that a-I exists. This property is satisfied by C(X), for any space X. It is well-known that a semi-prime f-ring A satisfies the bounded inversion property precisely when each maximal ideal A is an I-ideal. It then follows that the space Max(A) of maximal ideals of A, relative to its hull-kernel topology, is Hausdorff. (Max(A) is always compact; we refer the reader to [23, Lemma 0.0].)

Now, let bA stand for the (continuous) map which assigns to each minimal prime P the maximal idealbA(P) containing it; this map is well-defined because

60 A.w. HAGER AND JORGE MARTINEZ

the prime £-ideals of A forms a root-system (see [5]) which means that no two incomparable primes contain a third prime. t5 A is a continuous surjection of Min(A) on Max(A).

The details of this proof and most of the results which follow in the sequel will be omitted; they will appear in [16].

COROLLARY 1.3 Suppose that A is a semi-prime f-ring satisfying the bounded inversion property, and that J(A) = O. Then A is fraction-dense if and only if Min(A) and bA realize the absolute of Max (A).

Note that the class of semi-prime f-rings A with bounded inversion, which in addition satisfy J(A) = 0, includes the Archimedean f-rings. (See [23, Discussion preceding 3.9].) The converse is false, as evidenced by a non-Archimedean ordered field.

Before proceeding to examine fraction-dense spaces more closely, a comment is in order on the heels of Corollary 1.3. Let us assume that A stands for a semi-prime f-ring with the bounded inversion property.

Since QA is von Neumann, we have that Max(QA) = Min(QA). On the other hand, in QA every polar is principal, and the algebra is orthocomplete, which means Max( Q A) is extremally disconnected.

Next, if B is any f-subring of A (also with bounded inversion), consider the map B: Max(A) -+ Max(B) which assigns to a maximal ideal M of A the unique maximal ideal of B which contains the contraction M n B. As shown by Woodward, and soon to appear in his dissertation, this is a continuous surjection of Max(A) on Max(B). So let us consider this map BA:Max(QA) -+ Max(A).

PROPOSITION 1.4 The map BA: Max(QA) -+ Max(A) realizes the absolute of Max(A), for each semi-prime f-ring A which satisfies the bounded inversion property and J(A) = O.

Proposition 1.4 has the following appealing corollary. There are some details which need checking; we leave this to the reader as an exercise.

COROLLARY 1.5 Suppose that A is a semi-prime f-ring with bounded inversion and J (A) = O. Then the following are equivalent.

1) Max(A) is extremally disconnected. 2) A is strongly projectable. 3) A and QA have the same idempotents. 4) A is fraction-dense and bA is a homeomorphism from Min(A) onto Max(A).

We shall proceed now to a more detailed study of fraction-dense spaces. As far as we know every fraction-dense space is strongly fraction-dense; although it is unlikely that this implication holds in general, we have yet to discover a counterexample.


By contrast, observe that if A is the I-algebra of all real sequences with finite range, then qA = A, while QA is the algebra of all real sequences. A is rigid in QA, so that A is fraction-dense, but not strongly fraction-dense.

2. Fraction-Dense Spaces

We begin by recalling the following definitions: a space X is cozero-complemented if for each cozero-set U there is a cozero-set V which is disjoint from U and such that the union is dense in X. Then observe that, since any fraction-dense I -ring is complemented, every fraction-dense space is cozero-complemented.

Now let us recall the following notion from [19]: X is a cloz-space if every complemented cozero set has an open closure. It follows immediately that every fraction-dense cloz-space is basically disconnected; that is, every cozero has an open closure. (Note: every quasi-F space is a cloz-space, and for strongly zerodimensional spaces the converse is true. The reader is referred to [10] and [19]; we shall return to quasi-F spaces shortly.) With a little more work one gets:

PROPOSITION 2.1 Eachfraction-dense cloz-space X is extremally disconnected.

Proposition 2.1 serves as a cue to the introduction of the notion of covers of topological spaces. We shall state a corollary, which follows immediately from the proposition, and then proceed to a discussion of covers.

COROLLARY 2.2 For any fraction-dense space X, the cloz-cover and the absolute coincide.

We have already mentioned irreducible maps. Now recall that a continuous map is perfect if it is a closed mapping and the inverse image of every singleton set is compact; evidently, if the spaces in question are compact, then every continuous map between them is perfect. Now, if f: Y -+ X is a perfect, irreducible surjection, we say that Y is a cover of X. For a comprehensive discussion of the theory of covers we refer the reader to [15]. Much of the deep work on this subject has been done by Vermeer; see [27] and [28].

Let us consider the lattice COV(V). First, recall that two covering maps f: Y -+ X and g: Z -+ X are said to be equivalent, if there is a homemorphism h: Y -+ Z such that gh = f. Such a map is unique. Modulo this equivalence relation one can then order COV(X), the collection of all covers of X, as follows: with the same designations for I and 9 as before, we say that I 2: 9 if there is a continuous map h: Y -+ Z such that gh = f. We note that I 2: 9 and 9 2: I together imply that I and 9 are equivalent. (See [15] for details; if I 2: 9 then the map h is perfect and irreducible and unique with respect to making gh = I.) Then under this partial ordering COV(X) is a complete lattice, in which the least element is (X, 1 x ), while ( EX, ex) is the largest element. The suprema in COV(X) can be described by means of pullbacks; see [15].

62 A.w. HAGER AND JORGE MARTINEZ

Now suppose that K is a class of topological spaces. (One need not assume that the spaces are Tychonoff, but we shall continue to do so.) We say that K is a covering class if for each space X E K there is a least element (Y, J) (where f: Y ----7 X is perfect and irreducible) with Y E K. If such a minimum cover exists we speak of the K-cover of X, and denote it by KX.

Thus, if E is the class of extremally disconnected spaces, then every space X has a E-cover, namely EX, the absolute space. Let QF denote the class of quasi-F spaces; (X is quasi-F provided every dense cozero-set is C* -embedded.) Various contributions to the literature have discussed the quasi-F cover: [10], [18] and [20]. In [19] the authors show that every space X has a doz-cover, EccX.

This will suffice as a discussion of the highlights from the theory of covers. Since every cloz-space is quasi-F, it follows that for every fraction-dense

space X, QF X = EX. The converse is false: let X be the space of all ordinals less than the first uncountable ordinal, endowed with the order topology. By [18,3.15], EX = QFX; however, X is not fraction-dense: {lX is its one-point compactification, by adjoining the first uncountable ordinal, which is a P-point in {lX. It can be shown that if the cardinality of a compact fraction-dense space is non-measurable, then every P-point is isolated. Thus {lX is not fraction-dense.

On the other hand, from work of Ball-Hager, [3], we get:

PROPOSITION 2.3 If X is a compact space, then X is fraction-dense if and only ifQFX = EX.

In the remainder of this section we shall assume that all spaces are compact. We shall also assume that, henceforth, all f-rings satisfy the bounded inversion property.

Following [3], we shall say that the perfect, irreducible map f: Y ----7 X is a subsequentially irreducible if for each cozero-set W s;:; Y there is a cozero-set V s;:; X such that W and f- 1 (V) have the same closure. This notion is called Z* -irreducible in [18] and WI-irreducible elsewhere. As is demonstrated in [18, Thm. 2.13], the quasi-F cover (Y, ¢) of a space X is characterized by Y being a quasi-F space and the covering map ¢: Y ----7 X being sequentially irreducible.

From [3, Thm. 2.4] we conclude the following:

PROPOSITION 2.4 Suppose that f: Y ----7 X is a perfect, irreducible map. Then f is sequentially irreducible if and only if the induced embedding C (J): C (X) ----7

C (Y) is a rigid embedding.

We now summarize some of the above in a corollary.

COROLLARY 2.5 For a compact space X the following are equivalent. 1) X is fraction-dense. 2) QFX = EX. 3) The canonical map ex : EX ----7 X is sequentially irreducible.


We close this section by recording the following corollary.

COROLLARY 2.6 A compact space X is fraction-dense if and only if every regular open subset of X densely contains a cozero-set.

3. Coincidence of Completions

f -Rings with bounded inversion are divisible, as additive groups. In addition, conditions 5) and 6) of Theorem 1.1 suggest how to define 'fraction-density' for (Abelian) lattice-ordered groups. Assume that G is an arbitrary Abelian latticeordered group; we say that it is absolute if Min( G) is compact and extremally disconnected.

If G has an order unit u, let Yos( G, u) (or Yos( G), if the unit is fixed or otherwise understood) denote the set of values of u; that is, the set of all primes £ideals of G which are maximal with respect to excluding u. Yos( G) is a compact, Hausdorff space relative to its hull-kernel topology; as is customary (see [l3, Chapt. 8] or [3] for examples) we shall refer to this space as the Yosida space of G. Let us now examine the relationship between the absoluteness of G and covers of Yos(G).

Let us begin by recalling the notion of an (order) essential extension of a lattice-ordered group. Suppose that G is an £-subgroup of the lattice-ordered group H; then H is an essential extension of G if each non-trivial f-ideal of H has a non-trivial intersection with G. If Gis Archimedean then it has a (unique) maximal, Archimedean essential extension, denoted eG; see [7] for details. Here we add one more observation about eG: it is £-isomorphic to D(X), where X is a Stone-dual of PC G).

On the other hand, we observe the following; assume that G is a complemented lattice-ordered group. For each 9 E G the basic open set u(g), consisting of all the minimal primes of G which exclude g, is compact-open, and u(g) is homeomorphic to Min(g-Ll) = Min(G(g)), where G(g) denotes the f-ideal generated by g. Therefore, if G is absolute, then so is G(g), and u(g) is the absolute ofYos(G(g),g), while the map eg:u(g) = Min(G(g)) --+ Yos(G(g),g), assigning the minimal prime f-ideal P E u(g) to the value of gin G(g) containing P n G (g) realizes the absoluteness of u(g) and is sequentially irreducible. (The proofs mimic the ones for f-rings completely.) Thus:

PROPOSITION 3.1 Suppose that G is an Abelian fl.-group. Thenjor each g E G, Yos( G (g), g) is a fraction-dense space.

For the remainder of this article we suppose that G is a divisible Archimedean lattice-ordered group.

We recall the notion of o-convergence and the associated o-completion; (see [10], [20] or [25] for details.) We say that a sequence Cgn) in Go-converges to g if


there is a decreasing sequence (vn ) of positive elements such that infn Vn = 0 and Ilgn - gil::; Vn· The o-Cauchy condition is defined analogously, and we say that G is o-complete if every o-Cauchy sequence converges.

We shall not recall the precise definition of a-completion here, but only recollect that if X is any compact space then C(QFX) is the o-completion ([10, Thm. 3.9b]). Recall as well that C(EX) and C(BDX) are, respectively, the Dedekind and Dedekind o-completions of C(X). Clearly, X is fraction-dense if and only if these completions coincide. The next lemma generalizes this; as it is a crucial lemma, we give a proof in this instance.

LEMMA 3.2 Suppose that G is an Archimedean lattice-ordered group. If G is absolute and a-complete ,then it is Dedekind complete.

Proof Clearly G is Dedekind complete if and only if each G(g) is Dedekind complete. Since the absoluteness and the o-completeness of G imply the same for each G(g), we may assume without loss of generality that G has a strong order unit u.

The lohnson-Kist-Yosida embedding (see [22, Chapt. 7]) then puts G in C(Yos(G)); from Proposition 3.1 we have that Yos(G) is fraction-dense.

Now it is well-known that an o-complete £-group is uniformly complete. This observation, together with the one that G is uniformily dense in C(Yos(G))and we need the divisibility here - implies that G = C(Yos(G)). Since G is 0-

complete Yos(G) must be a quasi-F space ([10, Thm. 3.7]), which implies that Yos( G) is extremally disconnected, by Proposition 2.1, and hence G is Dedekind complete. Q.E.D.

We have the following theorem:

THEOREM 3.3 Suppose that G is an Archimedean lattice-ordered group with order unit. If G is absolute then its o-completion and Dedekind completion coincide. The converse is true provided Pre G) is o-complete.

One might be able to relax the a--completeness of Pre G) in Theorem 3.3; however, it cannot be discarded altogether. If A is the ! -ring of all eventually constant sequences of real numbers, then Pr(A) is not a--complete and A is not fraction-dense - not absolute, as a lattice-ordered group. However, the 0-

completion of A coincides with its Dedekind completion; namely, C(8N).

References

1. M. Anderson, P. Conrad, The hulls of C(X), Rocky Mountains J. 12:1 (Winter 1982), pp.7-22.

2. M. Anderson, T. Feil, Lattice-Ordered Groups; an Introduction, Reidel, Dordrecht, 1988. 3. B.H. Ball, A.w. Hager, Archimedean kernel-distinguishing extensions of Archimedean

£-groups with weak unit, Indian Jour. Math. 29:3 (1987), pp.351-368. 4. B. Banaschewski, Maximal rings of quotients of semi-simple commutative rings, Archiv.

Math. XVI (1965), pp. 414-420.


5. A. Bigard, K. Keime1, S. Wolfenstein, Groupes et Anneaux Reticules, Lecture Notes in Mathematics 608 , Springer-Verlag, Berlin, 1977.

6. R. Bleier, The orthocompletion of a lattice-ordered group, Proc. Kon. Ned. Akad. v. Wetensch., Ser A 79 (1976), pp. 1-7.

7. P. Conrad, The essential closure of an Archimedean lattice-ordered group, Proc. London Math. Soc. 38 (1971), pp. 151-160.

8. P. Conrad, J. Martinez, Complemented lattice-ordered groups, Proc. Kon. Ned. Akad. v. Wetensch. New Series 1 (1990), 281-297.

9. P. Conrad, J. Martinez, Complemented lattice-ordered groups, Order to appear. 10. F. Dashiell, A.W. Hager, M. Henriksen, Order-Cauchy completions of rings and vector

lattices of continuous functions, Canad. 1. Math. 32 (1980), pp. 657-685. 11. N. Fine, L. Gillman, J. Lambek, Rings of Quotients of Rings of Functions, McGill University,

1985. 12. A. Gleason, Projective topological spaces, lllinois 1. Math. 2 (1958), pp. 482-489. 13. L. Gillman, M. Henriksen, Rings of continuous functions in which every finitely generated

ideal is principal, Trans AMS 82 (1956), pp. 366-391. 14. L. Gillman, M. Jerison, Rings of Continuous Functions, Grad. Texts in Math. 43, Springer

Verlag, Berlin, 1976. 15. A.W. Hager, Minimal covers of topological spaces, Ann. NY Acad. Sci., Papers on Gen.

Topol. & ReI. Cat. Th. & Top. Alg. 552 (1989), pp. 44-59. 16. A.W. Hager, J. Martinez, Fraction-dense algebras and spaces, submitted. 17. M. Henriksen, M. Jerison, The space of minimal prime ideals of a commutative ring,

Trans. AMS 115 (1965), pp. 110-130. 18. M. Henriksen, J. Vermeer, R.G. Woods, Quasi-F covers of Tychonoff spaces, Trans

AMS 303:2 (Oct. 1987), pp. 779-803. 19. M. Henriksen, J. Vermeer, R.G. Woods, Wallman covers of compact spaces, Diss. Math.

to appear. 20. C.B. Huijsmans, B. de Pagter, Maximal d-ideals in a Riesz space, Canad. 1. Math.

XXXV:6 (1983), pp. 1010-1029. 21. J. Lambek, Lectures on Rings and Modules, Ginn-Blaisdell, Waltham Mass., 1966. 22. W.A.J. Luxemburg, A.C. Zaanen, Riesz Spaces, I, , North-Holland, Amsterdam, 1971. 23. J. Martinez, The maximal ring of quotients of an f -ring, submitted. 24. B. de Pagter, The space of extended orthomorphisms in a Riesz space, Pac. 1. Math. 112

(1984), pp. 193-210. 25. F. Papangelou, Order convergence and topological completion of commutative lattice

groups, Math. Ann. 55 (1964), pp.81-107. 26. J.R. Porter, R.G. Woods, Extensions and Absoluteness of Hausdorff Spaces, Springer

Verlag, Berlin, 1988. 27. J. Vermeer, On perfect irreducible preimages, Topology Proc. 9 (1984), pp.173-189. 28. J. Vermeer, The smallest basically disconnected preimage of a space, Topol. App!. 17

(1984), pp. 217-232. 29. R. Walker, The Stone-tech Compactification, Ergebnisse der Math. und ihre Grenzgeb. 83

Springer-Verlag, Berlin, 1974. 30. A.W. Wickstead, The injective hull of an Archimedean f -algebra, Compos. Math. 62

(1987), pp. 329-342.


An Alternative Proof of a Radon-Nikodym Theorem for Lattice Homomorphisms

C.B. HUnSMANS Department of Mathematics, University of Leiden, p.o. Box 9512, 2300 RA Leiden, The Netherlands.

and

W.AJ. LUXEMBURG Department of Mathematics, California Institute of Technology, Pasadena, California 91125, U.S.A.


Abstract. We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.

Mathematics Subject Classifications (1991): 46A40, 47B60

Key words: Radon-Nikodym type theorem, lattice homomorphism, vector lattice

67

It is the aim of this note to present a new proof of the following Radon-Nikodym type theorem for (linear) lattice homomorphisms, due to W.AJ. Luxemburg and A.R. Schep [4, Thm. 4.2]. Throughout E is an Archimedean and F a Dedekind complete vector lattice. We denote the Dedekind complete vector lattice of all order bounded (= regular) linear operators from E into F by Lb(E, F) and the subset of all lattice homomorphisms from E into F by Hom( E, F). In the next theorem the equivalence of i) and ii) is the above quoted Luxemburg-Schep result, whereas the equivalence with iii) or iv) is shown by C.B. Huijsmans and B. de Pagter in [3, Lemma 5.l]. If T E Hom(E, F) and S E Lb(E, F)+, then the following are equivalent:

i) S E {T}dd; ii) S is absolutely continuous with respect to T (Le., Su E {Tu}dd for all

u E E+). If, in addition, E is Dedekind complete, then each of these statements is equivalent to iii) or iv): iii) S(B) c {T(B)}dd for all bands B in E;

68 C.B. HUIJSMANS AND W.A.J. LUXEMBURG

iv) Su E {T {U }dd}dd for all u E E+. In the present paper we shall give an alternative proof of the equivalence of i)

and ii). Notice already that the proof of the implication i) ===? ii) is easy. Indeed, if 0::; S E {T}dd, then S 1\ nT r S implies (S 1\ nT)u r Su for all u E E+. It follows from 0 ::; (S 1\ nT)u ::; nTu that (S 1\ nT)u E {Tu}dd (n = 1,2, ... ) and consequently Su E {TU}dd for all u E E+.

An essential ingredient in our approach of the proof of ii) ===} i) is a result due to M. Meyer [5] (for an easier, representation-free proof we refer to S.J. Bernau [2]), which reads as follows. If E and F are Archimedean vector lattices and S E £b(E, F) is disjointness preserving (i.e., f.lg in E implies Sf .lSg in F), then ISI,S+,S- exist; ISI,S+,S- E Hom(E,F) and

(Su)+ = S+u, (Su)- = S-u,

ISIIfl = ISlf11 = IISlfl = ISfl (f E E).

We first present two simple lemmas which will be needed in the sequel.

LEMMA 0.1 If E, Fare Archimedean vector lattices and S, T E Hom(E, F), then S + T E Hom(E, F) if and only if Su 1\ Tv = 0 for all u, v E E+ with u 1\ v = o.

Proof Suppose that S + T E Hom(E, F) and u 1\ v = O. It follows from

0::; Su 1\ Tv ::; (S + T)u 1\ (S + T)v = 0

that Su 1\ Tv = O. Conversely, take u, v E E+ with u 1\ v = O. Then

together with

0::; (S + T)u 1\ (S + T)v ::; Su 1\ Sv + Su 1\ Tv + Tu 1\ Sv + Tu 1\ Tv

implies

(S + T)u 1\ (S + T)v = 0,

so S + T E Hom(E). Q.E.D.

LEMMA 0.2 If E and Fare Archimedean vector lattices, T E Hom(E, F) and 0::; S: E -+ F satisfies Su E {Tu}dd for all u E E+, then S E Hom(E, F). In particular, if 0 ::; S E {T}dd, then S E Hom(E, F).

RADON-NIKODYM THEOREM FOR LATIICE HOMOMORPHISMS 69

Proof If u, v E E+, U /\ v = 0, then

0::; (8u /\ nTu) /\ (8v /\ mTv) ::; (n + m)(Tu /\ Tv) = 0

yields

(8u /\ nTu) /\ (8v /\ mTv) = 0 (m,n EN).

Since 8u E {Tu}dd, 8v E {Tv}dd we have 8u/\nTu In 8u, 8v/\mTv 1m Tv and hence 8u /\ 8v = O. Q.E.D.

The following Proposition and Corollary play a crucial role in our proof of the Radon-Nikodym theorem for lattice homomorphisms.

PROPOSITION 0.3 If E is an Archimedean and F a Dedekind complete vector lattice, T E Hom(E, F) and 0 ::; 81,82 E {T}dd, then

(81 /\ 82 )u = 8 1u /\ 82u

for all u E E+. Proof Put 8i = 81 - 81 /\ 82, 8~ = 82 - 81 /\ 82 and 8' = 8i - 8i.

lt follows from 8i,8i E {T}dd that 18'1 E {T}dd, so 18'1 E Hom(E,F) by Lemma 0.2. This implies that 8' is an order bounded disjointness preserving operator. Furthermore, 8i /\ 8i = 0 gives 8i = (8')+ and 8i = (8')-. By the above-mentioned result of Meyer and Bernau we get 8iu /\ 8iu = 0 for all u E E+, i.e.,

and consequently

81u /\ 82u = (81 /\ 82)u

for all u E E+. Q.E.D.

REMARK 0.4 In general the result of Proposition 0.3 does not hold for the dual notion of a lattice homomorphism, viz. the interval preserving (or Maharam) operators. By way of example, take E = L I ([0, 1]), F = R, and define

Tf = 101 f(x) dx, 1

8d = 102 f(x) dx, 8d = 11 f(x) dx 2

for all fEE. Then 8 1, 82 and T, being positive linear functionals on E, are interval preserving. Moreover, 0 ::; 81,82 ::; T shows that 81,82 E {T}dd. If 1 denotes the function identically 1 on [0,1], then (81 /\ 82)(1) = 0 (hence 81 /\82 = 0), but 811/\ 821 = ~.

70 C.B. HUIJSMANS AND W.A.J. LUXEMBURG

COROLLARY 0.5 Let E be an Archimedean and F a Dedekind complete vector lattice. If 51,52 E Lb(E,F)+ and 51 + 52 E Hom(E,F) (so 51,52 E Hom(E, F)), then the following are equivalent.

i) 51 /\ 52 = 0; ii) 5 1u /\ 52u = o for all u E E+;

iii) 5 1u /\ 52v = 0 for all u, v E E+; iv) 51 (E)~52(E) (i.e., 5t1 ~52g for all j, gEE).

Proof The equivalence of ii), iii) and iv) and the implication ii) ===} i) are obvious.

i) ===} ii) Applying Proposition 0.3 to T = SI + S2, we get

for all u E E+.

Q.E.D.

REMARK 0.6 Another way of verifying i) ===} ii) above goes via a result of 5.5. Kutateladze (see [1, Thm. 8.16] for a simple proof). The same idea is used in [3, the proof of Lemma 5.1]. Indeed, assume that 51 + 52 E Hom(E, F) and 51/\52 = O. It follows from 0 ::; 51,52 ::; 51 +52 that there exist orthomorphisms 1l'1,1l'2 E Orth( F) (0 ::; 1l'I, 1l'2 ::; IF, the identity on F) such that

Observe that

Put PI = 1l'I -1l'I/\ 1l'2, P2 = 1l'2 -1l'I/\ 1l'2. Then 0 ::; PI, P2 ::; IF and PI/\ P2 = O. Moreover, 51 = PI (51 + 52) and 52 = pz(51 + 52). Since the infimum of orthomorphisms is pointwise on positive elements, we obtain

5 1u /\ 52u = Pl(51 + S2)U /\ PZ(SI + S2)U = (Pl/\ P2)(51 + S2)U = 0

for all u E E+.

Now we are able to show how Corollary 0.5 enables us to give an alternate proof of the Radon-Nikodym theorem for lattice homomorphisms.

THEOREM 0.7 Let E be an Archimedean and F a Dedekind complete vector lattice. Let T E Hom(E, F) and 5 E Lb(E, F)+. Then the following are equivalent.

i) 5 E {T}dd; ii) 5u E {Tu}dd for all u E E+.

RADON-NIKODYM THEOREM FOR LATTICE HOMOMORPHISMS 71

Proof The proof of i) ===} ii) is indicated before, so ii) ===} i) remains to be shown. By Lemma 0.2, 8 E Hom(E, F). We claim that 8u /\ Tv = 0 for all u,v E E+ with u /\ v = O. Indeed, Tu /\ Tv = 0, so {Tu}dd n {Tv}dd = {O} as well. Now the claim is immediate from 8u E {TU}dd and Tv E {Tv}dd. Applying Lemma 0.1 we obtain 8 +T E Hom(E, F). Decompose 8 = 8 1 + 82

according to the order direct sum

(so 0 ~ 8 1 E {T}dd and 82 /\ T = 0). By Corollary 0.5 (note that 0 ~ 52 + T ~ 8 + T implies 82 + T E Hom(E, F», 82u E {TU}d for all u E E+. On the other hand, it follows from 0 ~ 82u ~ 8u and 8u E {TU}dd that 82u E {TU}dd for all u E E+. Hence, 52u = 0 for all u E E+, showing that 52 = 0, 5 = 51 E {T}dd and the proof is complete. Q.E.D.

REMARK 0.8 The equivalence in Theorem 0.7 does not hold in general for the ideal AT generated by T E Hom(E.F), that is, if 0 ~ 8 E AT, then (as is easily checked) 8u E ATu for all u E E+, but not conversely. The following counterexample is due to A.W. Wickstead ([6, Sect. 4]).

Take E = F = (coo) and let 8 be a multiplication in (coo) by a positive unbounded sequence. It is straightforward to verify that 8u E Au for all u E E+, but that 8 rt. AI (where I is the identity mapping).

References

1. C.D. Aliprantis, D. Burkinshaw, Positive Operators, Acad. Press, Orlando, 1985. 2. S.l. Bemau, , Orthomorphisms of Archimedean vector lattices, Math. Proc. Cambridge

Phil. Soc. 89 (1981), pp. 119-128. 3. C.B. Huijsmans, B. de Pagter, Disjointness preserving and diffuse operators, Compositio

Math. 79 (1991), pp.351-374. 4. W.A.J. Luxemburg, A.R. Schep, A Radon-Nikodym type theorem for positive operators

and a dual, Indag. Math. 40 (= Proc. K.N.A. W. 81) (1978), pp. 357-375. 5. M. Meyer, Le stabilisateur d'un espace vectoriel reticule, C.R. A cad. Sci. Paris ser. A

283 (1976), pp. 249-250. 6. A.W. Wickstead, Representation and duality of multiplication operators on Archimedean

Riesz spaces, Compositio Math. 35 (1977), pp. 225-238.


Some Remarks on Disjointness Preserving Operators

C.B. HUIJSMANS Department of Mathematics, University of Leiden, p.o. Box 9512, 2300 RA Leiden, The Netherlands.

and

B. DE PAGTER

73

Delft University of Technology, Faculty of Technical Mathematics and Infonnatics, Department of Pure Mathematics, P.O. Box 5031, 2600 GA Delft, The Netherlands


Abstract. In this note we present a simple proof of the following results: if T: E -+ E is a lattice homomorphism on a Banach lattice E, then: i) !Y(T) = {1} implies T = I; and ii) reT - I) < 1 implies T E Z(E), the center of E.


Key words: lattice homomorphism, Banach lattice

In 1978 it was shown by W. Arendt, H.H. Schaefer and M. Wolff [15, Coroll. 2.2] that a lattice homomorphism T on a Banach lattice E for which the spectrum O"(T) = {I} is equal to the identity mapping Ion E. An elementary, though not short proof, due to the first author, can be found in [8], [9]. The result of [15] was extended by W. Arendt in [3, Corol. 3.6] who showed that if T E £(E) is disjointness preserving and O"(T) = {I}, then T = I. It is the aim of this note to present a simple and short proof of the latter result. We will use the same method of proof to show that a lattice homomorphism T on a Banach lattice E for which the spectral radius r(I - T) < 1 necessarily belongs to the center Z(E) of E. This result also occurs in [15, Lemma 3.3]. Generalisations can be found in [3, Thm. 3.5] and [4, Prop. 5.4].

For the basic theory of operators on Banach lattices and unexplained notions and terminology we refer to the standard monographs [2], [14], [19].

We recall some relevant notions. A (throughout linear) operator T on a Banach

74 C.B. HUlJSMANS AND B. DE PAmER

lattice E is said to be a lattice homomorphism whenever u /\ v = 0 in E implies Tu 1\ Tv = O. Furthermore, an operator T on E is called disjointness preserving if T f ~Tg for all f, gEE for which f ~g. It was shown by Y.A. Abramovich [1] (see also [12]) that a norm bounded disjointness preserving operator on E is automatically order bounded. If T is an order bounded and disjointness preserving operator on E then the modulus ITI exists, satisfies

ITIlfl = ITlfll = IITlfl = IT fl

for all fEE and ITI is a lattice homomorphism of E (see [11], [5]). Now assume that E is a Dedekind complete Banach lattice. Denote by Lb( E)

the Dedekind complete vector lattice of all order bounded operators on E. It is well-known that Lb(E) C L(E), the Banach algebra (with respect to the operator norm) of all norm bounded operators on E. Although in general Lb(E), equipped with the operator norm, is not closed in £(E), the space £b(E) is a Banach lattice algebra with respect to the regular norm IITlir = IIITIII. As usual, the order ideal in Lb(E) generated by I is denoted by Z(E), the center of E. It is well-known that Z(E) is a band in Lb(E) (cf. [19, Chapt. 20]). We denote the projection of Lb(E) onto Z(E) by V (for diagonal). Observe that V(ST) ;::: V(S)V(T) for all 0 ::; S, T E Lb(E). Trivially, V is contractive with respect to the regular norm, but surprisingly enough it was shown by J. Voigt in [18] that V is even contractive with respect to the operator norm.

We start with a simple observation.

LEMMA 0.1 Let E be a Dedekind complete Banach lattice and T E L( E) order continuous and disjointness preserving. /fV(T) is injective, then T E Z(E).

Proof. Decompose T = 7r + S according to the order direct sum Lb (E) =

Z(E)ffiZ(E)d (so 7r = V(T». It follows from ITI = 17r1+ISI that 17r1::; ITI and lSI::; ITI· Since ITI is an order continuous lattice homomorphism, ISI/\ I = 0 obviously implies (ITI'ISI) /\ ITI = O. Hence, (17r1'ISI) /\ ITI = 0, showing that 7rS~T. On the other hand, since 17r1 ::; ).J for some A > 0, we get

I7rSI ::; 17r1·ISI ::; 17r1·ITI ::; AITI·

Hence, 7rS = O. Since 7r is 1 - 1, we derive S = 0, so T = 7r E Z(E) and the proof is complete. Q.E.D.

The following proposition is the key of our considerations.

PROPOSITION 0.2 Let E be a Dedekind complete Banach lattice, T E L(E) order continuous and disjointness preserving such that II (T - I)nll < 1 for some n E N. Then Tn! E Z(E).

Proof. Since NJr = R~ for every 7r E Z(E) (where NJr is the kernel and RJr the range of 7r; cf. [19, Thm. 140.5]), it follows easily that NJr = NJrk

DISJOINTNESS PRESERVING OPERATORS 75

(k = 1,2, ... ). Moreover, Tk is disjointness preserving as well and ITkl =

ITlk(k = 1,2, ... ). By Lemma 0.1 it suffices to show that D(Tn!) is injective. To this end, pick

fEE for which D(Tn!)(f) = O. It follows from

V(ITln!)lfl = IV(ITn!I)(f)1 =

= IIV(Tn!)I(f)1 = IV(Tn!) (f) I = 0

that V(ITln!)lfl = 0 as well. Since for k = 1, ... ,n,

we have D(ITlk)'tlfl = 0, so by the remark at the beginning of the proof V(ITlk)lfl = 0 and hence V(Tk)(f) = 0 (k = 1, ... ,n).

The above quoted result of Voigt implies

i.e.,

Hence, I:~=1 G)(-l)n-k1)(Tk) is invertible in £(E). Since

we find f = 0, which is the desired result. Q.E.D.

We are now in a position to present a simple proof of the Schaefer-WolffArendt result.

THEOREM 0.3 Let E be a Banach lattice and T E £(E) disjointness preserving such that cr(T) = {I}. Then T = I.

Proof The operator T is invertible in £(E) as 0 t/:. cr(T). Hence, T* is disjointness preserving (see, e.g., [3, Prop. 2.7]). Since E* is Dedekind complete, T* is order continuous and cr(T*) = {I}, we may assume without loss of generality that E is Dedekind complete and that T is order continuous.

It follows from cr(T) = {I} that

lim II(T - I)kllt = r(T - I) = O. k---->oo

76 C.B. HUIJSMANS AND B. DE PAG1ER

Hence, II(T - 1)nll < 1 for some n EN. By Proposition 0.2, T m E Z(E) with m = nL It follows then from (J(Tm - 1) = {O} that

IITm - III = r(Tm - 1) = 0

and hence T m = I. Let WI = 1, W2, ... ,Wm be the complex m th roots of unit. Then

The terms T - w2I, . .. ,T - wmI are invertible in £(E), as W2, ... ,wm t/:. (J(T). Multiplying the last equality with the inverses we get T - I = 0, i.e., T = I and we are done. Q.E.D.

THEOREM 0.4 Let E be a Banach lattice, T: E --+ E a lattice homomorphism such that reT - 1) < 1. Then T E Z(E).

Proof Again, 0 tf. dT), so we may assume as above that E is Dedekind complete and that T is order continuous. Since

reT - 1) = lim II(T - 1) k ll* < 1 k __ =

there exists, by Proposition 0.2, a natural number m such that T m E Z(E). Moreover, (T-I)m = T-m exists in L(E), as T is invertible in L(E). But T m E Z(E), (Tm)-I E L(E) implies T-m E Z(E), as Z(E) is a full sub algebra of L(E). Noting that T m is a lattice isomorphism, we find T- m 2: O. Now Z(E) is an Archimedean uniformly complete i-algebra with unit element, so T-m has a unique positive mth root in Z(E), i.e., there exists a unique 0 :s; 'Jr E Z(E), commuting with T, such that 'Jrm = T- m (see [6], [13]). Since 'Jrm is invertible in Z(E), so is 'Jr. Put S = 'JrT, which is clearly a lattice isomorphism of E.

It follows from r( I - T) < 1 that

(J(T) C {A E C: ReA> O}.

Moreover, S = 'JrT = T'Jr implies (J(S) C (J('Jr)' (J(T), so (J('Jr) C [a, b] for some o < a :S b yields

(J(S) C {A E C : ReA> O}

as well. Furthermore,

sm = 'Jrm . T m = T-m . T m = I.

Hence

DISJOINTNESS PRESERVING OPERATORS 77

where Wk (l :::; k :::; m) denote the mth roots of unity in C. Therefore, if S :/= J, the cyclicity of O"(S) (a result due to E. Scheffold [16], based on theorems of H.P. Lotz [10]) would imply that

0"( S) n {A E C : Re). < O} :/= 0,

a contradiction. Hence, S = I and so T = 1T- 1 E Z(E) which finishes the proof. Q.E.D.

REMARK 0.5 In the proof of the last theorem we made an appeal to the rather deep result that the spectrum of a lattice homomorphism on a Banach lattice is cyclic. In this special case this can be shown more directly to the effect that if the operator S in the above proof (satisfying sm = I) has a spectral value w :/= 1, then w, w2, . .. ,wm E O"(S). Indeed, since S has a rational resolvent (use A.E. Taylor and D.C. Lay [17, Thm. 11.2 from Chapt. V]), every spectral value of 5 is a pole of the resolvent and hence an eigenvalue ([17, Sects. y'1Q and V.lI]), so w E O"p(5), the point spectrum of 5. There exists therefore lEE, I :/= 0, such that 51 = wi. As in the above proof we may assume that E is Dedekind complete, so there exists a bijective 0" E Z(E) such that

! = O"I!I, 10"1 = I.

Consider now the Luxemburg t-map associated to S as described in the dissertation of D. Hart [7], so t: Z(E) -+ Z(E) defined by t(1T) = 51TS- 1 for all 1T E Z(E). We follow the argument in this thesis, as presented in [7, proof of Thm. 5.1] (with c = 0). It follows from

that

51!1 = 15/1 = Iw!1 = Iwll!1 = III

wi = SI = 50"Ifi = t(0")5Ifl = t(O")I/I·

S(0"2Ifl) = t(O")S(O"I/I) = t(O")Sf = t(O")(wf) =

= t(O")(wO"I/I) = wt(O")(O"I/I) = wO"(t(O")lfl) =

= wO"(wf) = w20" f = W2(0"21/1)·

Since 0"21fl :/= 0, we derive w2 E O"p(S). Similarly it is shown that s(O"nl/l) = wn(O"nlfl), so O"nlfl :/= 0 yields wn E 0"(5)(n = 1, ... , m).

78 c.B. HUIJSMANS AND B. DE PAGTER

References

1. Y.A. Abramovich, Multiplicative representation of disjointness preserving operators, Indag. Math. 45 (1983), pp. 265-279.

2. CD. Aliprantis, O. Burkinshaw, Positive Operators, Acad. Press, Orlando, 1985. 3. W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. 1. 32 (1983),

pp. 199-215. 4. W. Arendt, D. Hart, The spectrum of quasi-invertible disjointness preserving operators,

1. Funct. Anal. 68 (1986), pp. 149-167. 5. S.l. Bernau, Orthomorphisms of arcbimedean vector lattices, Math. Proc. Camb. Phil.

Soc. 89 (1981), pp. 119-128. 6. F. Beukers, C.B. Huijsmans, Calculus in f-algebras, 1. Austr. Math. Soc. 37 (1984),

pp. 110-116. 7. D.R. Hart, Disjointness preserving operators, Dissertation, Pasadena, 1983. 8. C.B. Huijsmans, Elements with unit spectrum in a Banach lattice algebra, Indag. Math.

50 (1988), pp.43-51. 9. C.B. Huijsmans, An elementary proof of a theorem of Schaefer, Wolff and Arendt, Proc.

A.M.S. 105 (1989), pp. 632-635. 10. H.P. Lotz, Ueber das Spektrum positiver Operatoren, Math. Z. 108 (1968), pp. 15-32. 11. M. Meyer, Le stabilisateur d'un espace vectoriel reticule, C.R. Acad. Sci. Paris Ser. A

283 (1976), pp. 249-250. 12. B. de Pagter, A note on disjointness preserving operators, Proc. A. M.S. 90 (1984), pp.

543-549. 13. B. de Pagter, A functional calculus in f-algebras, Report 84-21 , Delft, (1984). 14. H.H. Schaeffer, Banach Lattices and Positive Operators, Grundlebren 215, Springer-Verlag,

Berlin, 1974. 15. H.H. Schaeffer, M. Wolff, W. Arendt, On lattice isomorphisms with positive real spectrum

and groups of positive operators, Math. Z. 164 (1978), pp. 115-123. 16. E. Scheffold, Ueber das Spektrum von Verbandsoperatoren in Banach verbanden, Math.

Z. 123 (1971), pp. 177-190. 17. A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, Wiley and Sons, New York,

1980 (2nd edition). 18. 1. Voigt, The projection onto the center of operators in a Banach lattice, Math. Z. 199

(1988), pp. 115-117. 19. A.C. Zaanen, Riesz spaces II, North-Holland, Amsterdam, 1983.


Weakly Compact Operators and Interpolation

LECH MALIGRANDA* Departamento de Matematicas, NIC Apartado 21827, Caracas 1020A, Venezuela


79

Abstract. The class of weakly compact operators is, as well as the class of compact operators, a fundamental operator ideal. They were investigated strongly in the last twenty years. In this survey, we have collected and ordered some of this (partly very new) knowledge. We have also included some comments, remarks and examples.

Mathematics Subject Classifications (1991): 47B07, 47A57

Key words: weakly compact operators, Banach space, interpolation

1. Introduction

The norm topology is too strong to allow any widely applicable subsequential extraction principles. Indeed, in order that each bounded sequence in X has a norm convergent subsequence, it is necessary and sufficient that X be finite dimensional. This fact leads us to consider another, weaker topology (weak topology) on normed linear spaces which is related to the linear structure of the spaces and to search for subsequential extraction principles therein. First of all, we will discuss the weak topology, weak compactness and reflexivity of Banach spaces. Then we would like to give an exposition of weakly compact operators between Banach spaces and Banach lattices. Almost an of this material is in principle available in some books.

Finally, in §5 there are considerations on interpolation of weakly compact operators from the recent work of Maligranda and Quevedo [20]. We also give some additional comments and remarks.

2. Weakly Compact Sets and Reflexivity

Let X be an infinite-dimensional Banach space. The weak topology O"(X, X*) is the weakest topology for which all bounded linear functionals on X are continuous, i.e., a net (xa) converges weakly to Xo if, for each x* E X*, lima x*(xa) = x*(xo).

The weak topology is linear (addition and scalar multiplication are continuous), Hausdorff (weak limits are unique) and not metrizable.

* Current address: Department of Mathematics, Lulea University, S-95187 Lu1ea, Sweden.

80 LEeR MALIGRANDA

EXAMPLE 2.1 (Neumann) Let A = {em + men: m < n, m, n E N} be a set in £2. Then 0 E Aweak, but there is no sequence in A which is weakly null. Therefore, the weak topology need not be metrizable.

Ryff [28] also constructed a subset of £1 which is bounded, weakly sequentially closed and not weakly closed. This example (as well as Neumann's) points out a norm closed set whose weak closure is obtained by adding a single point: the origin.

The weak topology is not complete, but some of the Banach spaces are sequentially weakly complete.

For any subset A c X we have All II c Aweak.

THEOREM 2.2 (Mazur) If A is a convex subset of a Banach space X, then All II = Aweak.

A subset A c X is said to be weakly compact if it is compact in the weak topology of X. A is said to be relatively weakly compact if the closure Aweak is weakly compact. A is said to be sequentially weakly compact if, for every sequence of elements of A, there is a subsequence which is weakly convergent to an element of A.

The map K-: X ---t X** defined by (K-x)(x*) = x*(x) for x EX, X* E X* is called the canonical embedding of X into X**. A Banach space X is said to be reflexive if K-(X) = X**.

Let A be a weakly compact set in X and x* E X*. Then by the BanachSteinhaus theorem x* is weakly continuous. Therefore x*(A) is a compact set of scalars and x* (A) is bounded for each x* E X*, i.e. A is bounded. Further, A is weakly compact, hence weakly closed, and so norm closed. Conclusion: weakly compact sets are norm closed and norm bounded. Fortunately, closed bounded sets need not be weakly compact.

Weakly compact sets in Banach spaces are plainly different from general compact Hausdorff spaces because they are sequentially compact, and each subset of a weakly compact set has a closure that is sequentially determined.

THEOREM 2.3 (Eberlein-Smulian) A subset of a Banach space X is relatively weakly compact if and only if it is relatively weakly sequentially compact. In particular, a subset of a Banach space is weakly compact if and only if it is sequentially weakly compact.

EXAMPLE 2.4 The unit ball of £1 is not weakly compact because by Schur's theorem in £1, weak and norm convergence of sequences coincide.

THEOREM 2.5 (Krein-SmuIian) If A is a weakly compact subset of a Banach space X, then conv(A) is also weakly compact.

WEAKLY COMPACT OPERATORS AND INTERPOLATION 81

From Theorem 2.3 we have some properties of weakly compact sets and the classical characterization of reflexivity.

THEOREM 2.6 Let X be a Banach space. Then the following are equivalent: i) X is reflexive;

ii) the unit ball of X is weakly compact; iii) the unit ball of X is sequentially weakly compact; iv) X* is reflexive.

Many interesting characterizations of reflexivity have been given by R.C. James. James (1951) supplied also a counter-example showing that the assumption /'1,( X) = X** cannot be replaced by X to be isometrically isomorphic to X**.

The classical Banach spaces playa central role in the development of general Banach space theory. How are the criteria for weak compactness or what does it mean for a sequence to converge weakly in these spaces? Let us give two examples.

EXAIV1PLE 2.7 Let f2 be any compact Hausdorff space. A bounded sequence in the space C(f2) is weakly convergent to zero if and only if it converges pointwise to zero.

EXAIV1PLE 2.8 (Dunford-Pettis) Let A be a bounded subset of Ll (f2, /L) with /Lf2 < 00. A set A is relatively weakly compact if and only if lim/-te-to SUPxEA

IIx1eliLl = O.

More about weakly compact sets can be found in Lindenstrauss's survey paper [18] and Floret's monograph [13].

3. Weakly Compact Operators and Factorization

A bounded linear operator T: X ~ Y between Banach spaces X and Y is weakly compact ifT(Bx) is relatively weakly compact. The weakly compact operators were used for the first time by S. Kakutani and K. Yosida in 1938.

Let us give some examples.

EXAMPLE 3.1 If either X or Y is reflexive Banach space then every bounded linear operator T: X ~ Y is weakly compact.

The proof follows from the fact that in a reflexive Banach space any bounded set is relatively weakly compact and from the following Banach-Dunford theorem: A linear map T: X ~ Y between the Banach spaces X and Y is norm-la-norm continuous if and only if T is weak-to-weak continuous.

82 LEeR MALIGRANDA

EXAMPLE 3.2 Compositions of a weakly compact operator with a bounded operator (or bounded with weakly compact) are weakly compact.

EXAMPLE 3.3 Let k E Loo([O, 1] x [0,1]) and 1 ::; p < 00. Then the operator K: L 1([0, 1]) ----t Lp([O, 1]) defined by

Kx(t) = 101 k(s, t)x(s) ds

is weakly compact. Indeed, for x E L1 and a measurable subset e of [0,1],

1

II(Kx)lellp::; (me)p sup Ik(s, t)lllxI11. s,tE[O,1j

Therefore, if 1 osuPllxlh911(Kx)leI11 = 0, and by the Dunford-Pettis criterion, K (E Ll) is relatively weakly compact in L 1 .

EXAMPLE 3.4 Let, for 1 ::; p < 00, Vp be the Banach space of continuous functions on [0, 1] with the finite p-variation

1

vp(X) ~ sup (~IX(tk) - x(tk-tlIP) • ,

where the supremum is taken over all partitions ° = to < t1 < ... < tn = 1 of [0,1]. The norm is defined by

Ilxllvp = Ilxllc + vp(x).

The embedding of Vp into C([O, 1]) is not weakly compact. Namely, let xn(t) =

~ - ~t if ° ::; t ::; ~ and xn(t) = ° if ~ ::; t ::; 1. Then Ilxnllvp = 1, {xn} is weakly Cauchy in C([O, 1]) and {xn } does not have a weakly convergent subsequence to a continuous function on [0,1].

Reflexive spaces playa central role for weakly compact operators because of an important factorization theorem. Before stating this theorem, a Davis-FigielJohnson-Pelczynski construction is needed.

Let X be a Banach space and let W be a convex, symmetric, norm bounded subset of X. For each n we put Un = W + 2-n Ex and denote by in the Minkowski functional of Un. For ° < e < 1 and 1 < p < 00 we set

Z ~ ZO;P ~ {x EX: IIxll· c (~(z-'njn(x))P); < 00 },

and let J: Z ----t X denote the natural inclusion. We refer to Z as the space of Davies, Figiel, Johnson, and Pelczynski, or DFJP space for short.


LEMMA 3.5 (DFJP, [11]) a) Z is a Banach space and] is continuous. b) J is Tauberian, i.e., x** E X**, J**x** E Y imply x** EX. c) Z is reflexive if and only if W is relatively weakly compact.

THEOREM 3.6 (Davis-Figiel-Johnson-Pelczynsky, [11]) Weakly compact operators factor through reflexive spaces, i.e., if T: X ---+ Y is a weakly compact operator, then there is a reflexive Banach space Z and bounded linear operators S: X ---+ Z, R: Z ---+ Y such that RS = T.

Proof Let T: X ---+ Y be weakly compact and let W = T(Bx). The operators J- 1 0 T: X ---+ Z and J: Z ---+ Y provide the required factorization. Q.E.D.

The above factorization theorem can be used to prove several standard results from antiquity.

THEOREM 3.7 (Gantmacher) Let T: X ---+ Y be a bounded linear operator between Banach spaces. Then the following are equivalent:

i) T is weakly compact; ii) T**(X**) C K:(Y);

iii) T* is weakly compact.

4. Weakly Compact Operators on Banach Lattices

A Banach lattice X is a Riesz space with a lattice norm. A Banach lattice is said to have order continuous norm whenever Xc>: t 0 implies Ilxc>: II t o. A Banach lattice X is a KB-space (Kantorovich-Banach space) whenever every increasing norm bounded sequence of X+ = {x EX: x ~ O} is norm convergent.

A subset A c X is said to be solid whenever Ixl ~ Iyl in X and yEA imply x E A. The solid hull of a set A is the smallest solid set that contains A and is precisely the set sol(A) := {x EX: ::3y E A with Ixl ~ Iyl}.

The main question here is:

QUESTION 1 Does a weakly compact operator between two Banach lattices factor through a reflexive Banach lattice?

Let us note that if in Lemma 3.5 X is a Banach lattice and W is also a solid set then the DFJP space Z is a Banach lattice. Therefore the answer to the main question will be 'yes' if sol(T(Bx)) is relatively weakly compact.

LEMMA 4.1 a) (Abramovich, [1]) If X is a KB-space and A is relatively weakly compact set, then sol(A) is relatively weakly compact.

b) (Wickstead, [30]) A Banach lattice X has order continuous norm if and only if for every relatively weakly compact subset A of x+ sol(A) is relatively weakly compact.

84 LEeR MALIGRANDA

EXAMPLE 4.2 (Meyer-Nieberg, [22]) In a Banach lattice with order continuous norm the solid hull of a relatively weakly compact set need not be relatively weakly compact. Let X = co(L 1([0,1])) and Xn = (x~) with x~ = sinnt if k ::; n and x~ = 0 if k > n. Then {Xn: n E N} is relatively weakly compact and {Ixnl} is not weakly convergent.

EXAMPLE 4.3 If A c Co is relatively weakly compact then sol(A) is relatively weakly compact.

Proof Since Co is an AM-space, the lattice operations are sequentially weakly continuous. In fact, by the Kakutani representation theorem every AM-space is a closed sublattice of some space of the type C(fl), where fl is compact. But Xn ---+ 0 weakly in C(O) if and only if (Xn) is norm bounded and Xn(t) ---+ 0 for every tEO. Thus, for example Xn ---+ 0 weakly implies IXnl ---+ 0 weakly. Therefore the set IAI = {Ial: a E A} is relatively weakly compact. From Lemma 4.1b) sol(IAI) is relatively weakly compact, but sol(IAI) = sol (A) and we are done. Q.E.D.

From Lemma 4.1 and Example 4.3 we have the following theorem which is mainly due to Aliprantis-Burkinshaw ([4]).

THEOREM 4.4 Factorization of weakly compact operators T: X ---+ Y through reflexive Banach lattice holds if either:

i) Y is a KB-space; ii) Y = Co;

iii) T is positive and Y has order continuous norm; iv) X* has order continuous norm; v) T has a factorization T = RS, where S: X ---+ Z and R: Z ---+ Yare weakly

compact and Z is a Banach lattice.

EXAMPLE 4.5 (Counterexample; Talagrand, [29]) There is a weakly compact positive operator T: £1 ---+ C([O, 1]) for which factorization through a reflexive Banach lattice does not hold.

In the proofs of the factorization Theorem 4.4iv) and v) it was important to know when the following property holds: (*) If S, T: X ---+ Y are positive operators, S ::; T and T is weakly compact,

then S is weakly compact. The answer was given by Aliprantis-Burkinshaw ([3]) and Wickstead ([30]).

THEOREM 4.6 a) Let X be a Banach lattice and let T: X ---+ X be a positive weakly compact operator. If an operator S: X ---+ X satisfies 0 ::; S ::; T, then S2 is a weakly compact operator.

b) Property (*) holds if and only if either X* or Y has an order continuous norm.


5. Interpolation of Weakly Compact Operators

Let X = (Xo, Xt} be a Banach pair, i.e., X o and Xl are Banach spaces continuously imbedded into a Hausdorff topological vector space. Define, as usual the spaces ~(X) = Xo n Xl and ~(X) = Xo + Xl with the norms Ilxlb = max(llxllxo' Ilxllxl) and Ilxll~ = K(l,x), where for t > 0,

K(t, x) = inf{llxollxo + tllxlllxl : x = Xo + Xl, X o E X o, Xl E Xd·

For ° < () < 1 and 1 ::::; p ::::; 00, the Lions-Peetre interpolation space Xo,p = (Xo, XI)o,p is the space of all X E ~(X) for which the norm

1

Ilxllo,p = (2:(2-0nK(2n,X))P)P nEZ

is finite. Let us note that Mn(X) ::::; K(2n,x) ::::; 2Mn(X), where Mn is the Minkowski

functional of the set Vn = Bxo +2-nBX1 (BXi is the unit ball of Xi,i = 0, 1), i.e., fLn(X) = inf{a > 0: x E aVn}. Hence Xo,p consists of those x E ~(X) for which the norm

1

Ilxll~,p = (2: (2-0nMn(x))p) P nEZ

is finite. If in the above definitions the sums are only over natural n, then we .. -+ - -+.

have the defimtIOn of the space Xo,p" Always Xo,p c Xo,p; If X o C Xl> then - -+ Xo,p = Xo,p·

Now let X = (Xo, Xl) and Y = (Yo, Yt) be two Banach pairs. We say that a linear operator T is bounded from the pair X into the pair Y, and write T: X -----7 Y, if T: ~(X) -----7 ~(Y) is a bounded linear operator and the restriction of T to the space Xi is a bounded operator from Xi into Yi, i = 0, l.

From the construction of the Lions-Peetre spaces we have that if T: X -----7 Y, then T: X o,p -----7 Yo,p is bounded and

IITllx e,p-->Ye,p ::::; IITllt~yo IITII~l-->Yi . The following result can be regarded as a generalization of the theorem of

Beauzamy [6] about the reflexivity of X O,p.

THEOREM 5.1 (Maligranda-Quevedo, [20]) Let 1 < p < 00 and T: X -----7 Y. Then T: Xo,p -----7 Yo,p is weakly compact if and only if T: ~(X) -----7 ~(Y) is weakly compact.

LEMMA 5.2 (M-Q, [20]) a) Let T: ~(X) -----7 Y be a continuous operator. Then the following are equivalent:

86 LEeR MALIGRANDA

i) T: .6.(X) -+ Y is weakly compact; ii) T: (.6.(X, L;(X))B,p -+ Y is weakly compact, for all 0 < () < 1 and all

1 < p < 00;

iii) T: X B,p -+ Y is weakly compact, for all 0 < e < 1 and all 1 < p < 00.

b) Let 1 < p < 00 and let T: X -+ YB,p be a continuous operator. Then T is weakly compact if and only if T: X -+ L;(Y) is weakly compact.

Proof a) i)::::} ii) The interpolation space

Z = (.6. (X) , L;(X))B,p

is a DFJP space constructed from X = I;(X) and W = I(B!1(X))' where I is the inclusion I: .6.(X) -+ L;(X). Since T(W) is relatively weakly compact in Y, it is possible to prove that To J: X -+ Y is weakly compact.

ii)::::} iii). The interpolation space X B,p is not, in general, a DFJP space. But is proved in [19] that it is imbeddable into Z = (.6. (X) , L;(X))B,p, i.e., if -!- ::; e < 1, then

Z = (L;(X), .6.(X))I-B,p = XB,p + XI-B,p

and so XB,p and Xl-B,p are in Z. That iii) implies i) is obvious, because X B,p => .6.(X). b) The proof of this part follows from the fact that if 1 < p < 00, then the

imbedding J: Y B,p -+ L; (Y) is a Tauberian operator, and from the characterization of Tauberian operators by relatively weakly compact sets given by Kalton and Wilansky (cf. [15, Thm. 3.2] and [24, Thm. 1.4]). Q.E.D.

Let us note that Lemma 5.2a) is also true for compact, weakly precompact (=Rosenthal) and Banach-Saks operators. Moreover, the equivalence between i) and ii) is true even for surjective closed operator ideals (see [14, Prop. 1.7]). Lemma 5.2b) and Theorem 5.1 are also true for weakly precompact operators. Theorem 5.1 holds for such classes of operators which are preserved by Lemma 5.2a) and Tauberian operators.

If in our Theorem 5.1 the operator is taken to be the identity, then we have the result of Beauzamy.

COROLLARY 5.3 (Beauzamy, [6]) Let 0 < e < 1 and 1 < p < 00. The space X B,p is reflexive if and only if the pair X is weakly compact, i.e., the imbedding I: .6.(X) -+ L;(X) is weakly compact.

Another proof of this result is the following. If the imbedding I is weakly compact, then the DFJP space (.6.(X) , L;(X)h

2'P is reflexive. But from [19] we have that (.6. (X) , L;(X)h p = Xl p and from the reiteration theorem from the complex

2' 2' interpolation space (see [7, Thm. 4.7.2]):

XB,p = [XBo,p,X!,plB,


for some eo and e1 in (0,1). Now, using the above and Calder6n's result (if either Xo or Xl is a reflexive space, then the complex interpolation space [Xo, Xdo is also reflexive, see [9, Thm. 12.2]) we have that X O,p is reflexive.

Let us note that in the nontrivial case, i.e., when the imbedding I is nonclosed, the space XO,l is nonreflexive. Indeed, by Levy's [17] result it contains an isomorphic copy of £1. However, X 0,1 is reflexive if and only if I is weakly compact and closed, which means that XO,l = ~(X) and ~(X) is reflexive (cf. [27]).

There are simple counterexamples showing that Theorem 5.1 (even Corollary 5.3) is not true for the complex method of interpolation.

EXAMPLE 5.4 (Mali granda, [19]) For 1 < p < 00 let us consider the Marcinkiewicz space

and the closed subspace L~oo which is the closure of Loo in LPoo. For 0< e < 1 and 1 < p < q < 00, let ~ = (1 - e)/p + e/q. Then the pair {aOO,L~OO} is weakly compact because L~oo cUe L~oo and [L~OO, aOO]o = L~oo are not reflexive spaces.

EXAMPLE 5.5 Denote by F LIthe Fourier transforms of functions in L 1 (0, 2 7r), i.e., the space of sequences (an)::Ooo such that 2::::"'00 aneinx E L1 (0, 27r) with the norm

The pair FLf = {FL 1(28n),FL1(2(8-1)n)} is weakly compact because £2 is an interpolation space (more precisely, the Gustavsson-Peetre or Ovchinnikov method applied to this pair gives £2, see [26]). Moreover, the complex method for this pair gives [FL 1(20n),FL1(2(0-1)n)]0 = FL1 (see [26]) and the space F L1 is not reflexive. The techniques used in Theorem 5.1 also work for the

more general real method of interpolation (cf. [21], [27]). It is also possible to formulate and prove a theorem for arbitrary interpolation functors ('abstract nonsense').

THEOREM 5.6 Let T: X -+ Y be such that T: .6.(X) -+ "E(Y) is weakly compact. Assume that F is an interpolation junctor such that:

i) for every E > ° there exists t > ° such that BF(X) C tBl1(X) +EB~(X);

ii) the imbedding J: F(Y) -+ "E(Y) is a Tauberian operator. Then T: F(X) -+ F(Y) is weakly compact.

88 LEeR MALIGRANDA

References

1. Yu. A. Abramovich, Weakly compact sets in topological K-spaces, Teor. Funkcii Funkcional. Anal. i Prilozen. 15 (1972), pp. 27-35 (in Russian).

2. M. H. Aizenshtein, Duality of interpolation functors, Studies in the Theory of Functions of Several Variables Yaroslavl (1986), pp. 3-11 (in Russian).

3. C. D. Aliprantis, O. Burkinshaw, On weakly compact operators on Banach lattices, Proc. Amer. Math. Soc. 83 (1981), pp. 573-578.

4. C. D. Aliprantis, O. Burkinshaw, Factoring compact and weakly compact operators through reflexive Banach lattices, Trans. Amer. Math Soc. 283 (1984), pp. 369-381.

5. C. D. Aliprantis, O. Burkinshaw, Positive Operators, Acad. Press, New York, 1985. 6. B. Beauzamy, Espaces d'lnterpolation Reels: Topologie et Geometrie, Lecture Notes in

Math. 666, Springer-Verlag, Berlin-Heidelberg-New York, 1976. 7. 1. Bergh, J. Lofstrom, Interpolation Spaces. An introduction, Springer-Verlag, Berlin

Heidelberg-New York, 1976. 8. Yu. A. Brudnyi, N. YA. Krugljak, Interpolation Functors and Interpolation Spaces I

North-Holland, Amsterdam:(to appear ). 9. A. P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math

24 (1964), pp. 133-190. 10. M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Haifa

1990 (manuscript). 11. W.J. Davies, T. Figiel, W.B. Johnson, A. Pe1czynski, Factoring weakly compact operators,

1. Functional Anal. 17 (1974), pp. 311-327. 12. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, Berlin-Heidelberg

New York, 1984. 13. K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer-Verlag, Berlin

Heidelberg-New York, 1980. 14. B. Heinrich, Closed operator ideals and interpolation, 1. Functional Anal. 35 (1980),

pp. 397-411. 15. N.J. Kalton, A. Wilansky, Tauberian operators on Banach spaces, Proc. Amer. Math. Soc.

57 (1976), pp. 251-255. 16. S.G. Krein, Yu. I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka,

Moscow, 1978; English trans!.: AMS, Providence, 1982. 17. M. Levy, L'espace d'interpolation reel (Ao, Al )e,p contient £p, C.R. Acad. Sci. Paris 289

(1979), pp. 675-677. 18. J. Lindenstrauss, Weakly compact sets - their topological properties and the Banach spaces

they generate, in: Proc. Symp. on Infinite Dimensional Topology, Annals of Math Studies 69 ,Princeton Univ. Press, 1972, pp. 235-273.

19. L. Maligranda, Interpolation between sum and intersection of Banach spaces, 1. Approx. Theory 47 (1986), pp. 42-53.

20. L. Maligranda, A. Quevedo, Interpolation of weakly compact operators, Arch. Math. 55 (1990), pp. 280-284.

21. M. Mastylo, On interpolation of weakly compact operators, Hokkaido Math. 1. (to appear (1991 ?».

22. P. Meyer-Nieberg, Zur schwachen kompaktheit in Banachverbanden, Math. Z. 134 (1973), pp. 303-315.

23. RD. Neidinger, Properties of Tauberian operators on Banach spaces, PhD. Dissertation, Univ. of Texas at Austin (1984), 144 pp.

24. RD. Neidinger, Factoring operators through hereditarily-£p spaces, Lecture Notes in Math. 1166, Springer-Verlag, Berlin-Heidelberg-New York, 1985, pp. 116-128.

25. R.D. Neidinger, Concepts in the real interpolation of Banach spaces, Longhorn Notes, The Univ. of Texas at Austin, Functional Analysis Seminar 1986-87 pp. 1-15.

26. v.I. Ovchinnikov, The Method of Orbits in Interpolation, Math. Reports 1 Part 2, Harwood Academic Publishers, 1984,349-516.

27. A. Quevedo, Some remarks on the reflexivity of the real interpolation spaces, 1. Math. Anal. Appl. (to appear).


28. J.V. Ryff, The weak closure of a certain set in £1, Amer. Math. Monthly 81 (1974), pp. 69-70.

29. M. Talagrand, Some weakly compact operators between Banach lattices do not factor through reflexive Banach lattices, Proc. Amer. Math. Soc. 96 (1986), pp. 95-102.

30. A.W. Wickstead, Extremal structure of cones of operators, Quart. 1. Math. Oxford Ser. (2) 32 (1981), pp. 239-253.


Aspects of Local Spectral Theory for Positive Operators

Dedicated to G. Maltese on the occasion of his 60th birthday

PETER MEYER-NIEBERG

91

Fachbereich Mathematikllnformatik, Universitiit Osnabriick D-4500 Osnabriick, Bundesrepublik Deutschland


Abstract. In this paper we will discuss the local spectral behaviour of a closed, densely defined, linear operator on a Banach space. In particular, we are interested in closed, positive, linear operators, defined on an order dense ideal of a Banach lattice. Moreover, for positive, bounded, linear operators we will treat interpolation properties by means of duality.

Mathematics Subject Classifications (1991): 47B65, 47All

Key words: Banach space, operator equation, linear operator, spectral theory

1. Introduction

This paper is mainly concerned with the following problem. Let E be a Banach lattice, or a function space, T: E -+ E positive, linear A > 0 and 0 ::; y E E. Does there exist a positive solution of the equation (AI - T)x = y? If E is a Banach lattice, and A > r(T), then it is well-known that the resolvent R(A, T) is a positive, linear operator. Consequently, x = R(A, T)y is the desired solution. We will consider some generalizations of this classical situation. In particular, we are interested in the following problems.

i) What happens if A ::; r(T). ii) What happens if we replace the norm of E by a non-equivalent one.

iii) What happens if the domain of T is a dense ideal J c E. Throughout this paper, let E, F be complex Banach lattices, and X a complex

Banach space. Let £(X) be the algebra of all bounded, linear operators T: X -+

X. For every T E £(X) let r(T) be the spectral radius, dT) the spectrum, and p(T) the resolvent set of T. If A E p(T), then R(A, T) denotes the resolvent. Moreover, let £(E)+ be the cone of all positive, linear operators T: E -+ E. For unexplained terminology concerning Riesz spaces or Banach lattices we refer to

92 PETER MEYER-NIEBERG

[4], [5] and [3]. For the remainder of this paper we consider the following situations.

1) T: D ---+ X is a linear operator, defined on a linear subspace D c X; or 2) T: J ---+ E is a positive, linear operator defined on an ideal J c E.

Moreover, in these two cases, let

Doo = {x ED: Tkx E D for all kEN}

and

J 00 = {x E J: Tk x E J for all kEN} .

For every 0 cI a E C we consider

U(a) = {x E Doo: ~ a-n-1Tnx unconditionally convergent} .

For all x E E, 0 cI a E C, let

IIxll(Q) ~ sup {L~ a-n-l1mxll : N c No finite},

where No = N U {O}.

REMARK 1.1 i) x E U(a) implies that Iixll(a) < 00.

ii) T Doo c Doo . iii) If TIDoo : Doo ---+ Doo is closed, then it is continuous with respect to the

graph norm. Proof Assertions i) and ii) are trivial. iii) easily follows from the closed graph theorem. Q.E.D.

PROPOSITION 1.2 Assume that T: D ---+ X is a closed, linear operator. For all a E C with a cI 0 it follows that (U(a), II . II(a)) is a Banach space such that TU(a) C U(a). Moreover, IITU(aJII :s; lal, and Ilxll :s; lalllxll(a) for each x E U(a).

The operator R(a): U(a) ---+ X, defined by

00

R(a)x = L a-n-1Tnx, n=O

is of norm :s; 1 and satisfies

(aI - T)R(a)x = R(a)(AI - T)x = x

for every x E U(a).

LOCAL SPECTRAL THEORY FOR POSITIVE OPERATORS

Proof. Clearly, U(a) is a T-invariant subspace of Doo such that

IITU(a) II :::; 1001

and

for all x E U(a).

93

To show the completeness of U(a) , let (Yk)'t' c U(a) be a II· II-Cauchy sequence. There is Y E X such that IIYk - yll -+ ° as k -+ 00. For E > ° there exists pEN such that IIYm - Yqll(a) :::; E for all m, q 2: p. Since T is closed, we conclude from

IITkYm - Tkypll :::; IIYm - Ypll(a)lalk+l

for all kEN that Y E Doo and

IITkYm - TkYl1 -+ ° as m -+ (X) for every kEN.

If N c No is finite, then

II L a-n-lTn(Ym - Yq)11 ::; E

nEN

for all m, q E N with m, q 2: p. Letting q -+ (X) it follows that

II L a-n-lTn(Ym - y) II ::; E

nEN

for all m 2: q. Hence Y E U(a) and IIYm - yll(a) -+ ° as m -+ 00. Q.E.D.

EXAI\1PLE 1.3 i) Let X = e[O, 1] and TI = I' for all I E D = el[O, 1]. It is clear that Doo and U(a) (a f= 0) contain all polynomials. On the other hand, by means of the Arzela-Ascoli theorem one can show that there does not exist any T-invariant subspace Do C Doo such that the restriction of T to Do is closed, and Do is infinite dimensional.

ii) For the sake of simpler formulas we will consider sequence spaces over N2 = {2, 3, ... }. Let

E=.eI,

Ii' ~ {x: IlxllF ... E n-1lxnl < CX+

G = { x: IlxIIG = E nlxnl < (X) } •

It is trivial that E, F, and G are Banach lattices such that GeE c F as ideals. We define the operator T and T :

T( )00 ()oo h {O ifkf=n2, Xn 2 = Yk 2 , were Yk = . k 2 Xn If = n , T(Xn)f = (zk)f with Zk = Xk2.

We consider the operator T on the different spaces E, F, and G.


i) E is T-invariant, and T is an isometry on E. ii) F is T-invariant, and T is compact and quasinilpotent on F.

iii) G fails to be T -invariant. Let D = {x E G: Tx E G}. It is easy to see that D contains the natural basis {en: n E N2}, the restriction of T to D is closed, and D is dense in G. However, U(a) = 0 for each 0 oJ a E C. Indeed, let i E N2 . By induction it follows that

II'Feillc = i 2n for all n E N. _ Consequently ei ~ U(a). Considering the dual operator T one can easily show that ei E U(a) for every i E N2 and 0 oJ a E C. Hence, U(a) is order dense in G', where

G' = {(Yn)2' : (nYn):::'=2 E ROO}. For every x E Doo we define

rT(x) = lim sup IITnxlll/n E [0,00], n ..... oo

the local spectral radius of T at x.

REMARK 1.4 For all x E Doo and 0 oJ a E C the following assertions hold. i) rT(x) < lal implies x E U(a).

ii) x E U(a) implies rT(x) ::; lal.

COROLLARY 1.5 Assume that T: D -+ X is closed, linear and 0 oJ a, f3 E C such that lal < 1f31. [tfollows that U(a) c U(f3). For every x E U(a) we have

Ilxll«(3) :S 1f311~llaIIIXII(a). Proof We fix x E U (a). For every finite subset N C No it follows that

2: (3-n- 1T nx ::; lal 2: f3-n- 1T nx nEN nEN (a)

::; lal 2: 1f3I-n- 1 1Irnx ll(a) nEN

00

n=O

This completes the proof. Q.E.D.

COROLLARY 1.6 Assume that T: D -+ X is closed, and x E Doo such that rT( x) < 00. For all A, f3 E C with I AI, 1f31 > rT( x) it follows that

rT (R(A)X) :S rT(x) ,

LOCAL SPECTRAL TIIEORY FOR POSITIVE OPERATORS 95

R({3)x - R()..)x = ().. - (3)R()..)R((3)x.

Moreover, the mapping).. -+ R()..)x is holomorphic on {)..: 1)..1 > rT(x)}. If, in addition, T is continuous, then rT(x) = rT(R()..)x). Proof We fix 0: E R such that 1)..1, 1{31 > 0: > rT(x). Let To be the restriction

of T to U(o:). Since IITol1 ::; 0:, it follows that

00

R().., To) = I:).. -n-1Ton, n=O

R()..)x = R().., To)x,

R()")U(o:) c U(o:).

And from

IITn R()..)xll ::; 0: IITn R()..)xll a

::; 0: IIR().., To)IIIITnxlla ::; o:n+l IIR()", To)llllxlla

we conclude that rT(R()..)x) ::; 0: for all 0: > rT(x). Moreover, if T is continuous, then it follows for every n E N that

IITnxl1 ::; II)J - TllllTn R(>,)xll·

This implies rT(x) ::; rT(R(>.)x). Hence, the local resolvent equation follows from the usual resolvent equation.

Q.E.D.

COROLLARY 1.7 i) For all T E LeX) there exists x E X such that rT(x) = reT).

ii) 1fT E £(E)+, then there exists Z E E+ such that rT(z) = reT). Proof i) We fix ).. E <J(T) with 1)..1 = reT). If the assertion fails to be

true, then rT(x) < 1)..1 for all x E X. We achieve U()") = X. Moreover, R()..) is continuous with respect to the original norm. This is a contradiction. Hence rT(x) 2: 1)..1 for at least one x E X. But of course, rT(x) ::; reT).

ii) Let x E E be as in i). If z = lxi, then the assertion follows immediately. Q.E.D.

THEOREM 1.8 Let T: J -+ E be positive, closed, and linear, ).. E C, and Olex E E+

i) If x E J oo and>' E R with>. > rT(x), then there exists Z E E+ such that ()"I - T)z = x.


ii) If there exists Z E J+ such that (AI - T)z = x, then x E Jex>' and A is real with A 2: rTeX).

Proof i) Since A > rT(x), the series ex>

R(A)x = L A-n-1Tnx n=O

is absolutely convergent. Consequently n

(AI - T) L A-j-1Tjx = x - A-n-1Tn+x -+ x j=o

as n -+ 00. Since T is closed, it follows that

0::; R(A)X E Jrx)) and (AI - T)R(A)X = x.

ii) From z -I 0 and 0 = Imx = Im()"I - T)z = (Im)..)z we conclude that A E R. It easily follows that AZ = Tz + x -I 0 and Tz + x 2: O. Hence).. > O. It follows from

)"z = Tz +x 2: 0 and

that x E J, and Tx E J. By induction, Tn z , Tnx E J for all n E N. Thus z, X E J ex>, satisfying

z 2: z - ).. -n-1Tn+1 Z

n 00

= L)..-j-I()..I -T)Tjz = L)..-j-1Tjx j=O j=O

for all n E N. Consider a > A. From 0 ::; A-j-1Tjx ::; z it follows that the series

00 00 ()..)J+l La-j-1Tjx = L - )..-j-1Tjx j=O j=O a

is absolutely convergent. Consequently, rT(x) ::; a for all a>)... Q.E.D.

COROLLARY 1.9 Let T: J -+ E be positive, closed, linear. If there is ).. E C such that R(A, T) E £(E)+, then T E £(E)+, and)" > reT).

Proof The previous Theorem implies that x E Jex> and A > rT(x) for all x E E+. Thus Joo = E, T E £(E)+, and)" > reT). Q.E.D.

Let T: J -+ E be positive, linear. For all A E R with A > 0 we define

J(A) = {e E E: Ixl E U(A)},

/lXI/A = 1/ Ixl I/CA) for all x E JeA).

Hence x E J(A), if and only if the series :L~o A-n-1Tnlxl is unconditionally convergent. The proof of the following Proposition almost is the same as the proof of Proposition 1.2.

LOCAL SPECTRAL THEORY FOR POSITIVE OPERATORS 97

PROPOSITION 1.10 1fT: J ---t E is a closed, positive, linear operator and A> 0, then (J(A), 11·11).) is a Banach lattice, satisfying IITAII :::; A andT J(A) C J(A). Here, T). = TI J(A)'

THEOREM 1.11 For T E .c(E)+ and A> ° the following assertions are equivalent.

i) r(TA) < A. ii) J(A) = J(a) for some ° < a < A.

iii) On J(A) the norms II· II and II· II). are equivalent. iv) R(A)J(A) C J(A). v) J(A) is closed in (E, II . II).

For the proof of this Theorem see [3, Sect. 4.4]. It is an open problem, whether or not this theorem holds in a similar form for a closed, positive, linear operator T: J ---t E. Now we will treat some interpolation properties of positive linear operators. The proof of the first theorem can be found in [3, Sect. 4.4].

THEOREM 1.12 Suppose that E and F are Banach lattices such that FeE. Let T E .c(E)+, satisfying TF C F and

a = r(TIF) < f3 = r(TIE)'

Assume that one of the following conditions is satisfied. i) F is a dense ideal in E.

ii) F is an order dense ideal in E, E has the Fatou property, and T is order continuous.

For all A with a < A < f3 we have

Fe J(A) c E and

THEOREM 1.13 Let E be a Dedekind complete Banach lattice and T E £(E)+. Suppose that there is a T'-invariant ideal G C E~ such that G is a Banach lattice with respect to some norm II . IIG, and G is norm determinating for E. If a = r(Tb) < f3 = r(TE), then for all A E (a, f3] there is a Banach lattice F(A), containing E as a dense ideal, which is T-invariant such that r = (TF()')) = A.

Moreover, [a, f3l C O'(TE), and the mapping A f----t F(A) is strictly decreasing on (a, f3].

Proof First we will construct aT-invariant Banach lattice F, containing E as a dense ideal, such that r(TF) :::; a.

It follows from Nakano's theorem, see [3, 1.4.14], that E is an ideal in G~ =

G'. We denote by F the II . IIG,-closure of E in G'. Clearly, E is a dense ideal in F and F is T-invariant. By the construction, TF is the restriction of (Tb), to the closed ideal F. This implies that r(TF) :::; r(Tb) = a.


In E' we consider the maximal chain of invariant ideals J()", E'):).. > O. If ).. E (a, J3j, then G c J().., E'). Now, for all x E F we define a numerical lattice seminorm

Ilxll:\' = sup{(x', Ixl) : x' E ball(J()", E'))+}.

From Ilx'lh S )..-lllx'il we conclude that ball(J()..,E')) C )..ball(E'). Thus, for all 0 :I x E E we have

o :lllxll~ s )..llxll·

Let H = {x E F: Ilxll~ < oo}. It is clear that II'II~ is a lattice norm on H. A simple proof, using some standard arguments, will show that (H, II . II:\,) is a Banach lattice. Moreover, E is an ideal in H and H is an ideal in F. If F()..) be the closure of E in H, then E is an ideal in F()..). For all x E F()")+ it follows that

IITxll:\' = sup{(x', Tx) : x' E ball(J()..,E'))+}

= sup{(T'x', x) : x' E ball(J()..,E'))+}::; )..llxll:\'.

For all x E E+ and n E N, let

n

x(n) = I:)..-j-1 Tjx. j=O

It follows that

Ilx(n) II:\' = sup { (x', x(n)) : x' E ball(J().., E'))+}

~ 'up { (E ;.-;-1 y';x', x) : x' E baU(.T(;', 8))+ }

::;sup { f)..-j-1 T'jx' IlxllE: x' Eball(J()..,E'))+} )=1 E'

::; IlxiIE.

Consequently, for all 0 ::; x' E J().., E') we have

00

Ilx'lh = II I: )..-j-l T'jx'IIE' j=O

~ sup { (t, ;.-;-IT';x', x) : n E N, x E baU(E) l }

= sup{(x',x(n)): n E N, x E ball(E)+}

::; sup{(x',z): Z Eball(F()..))+}.

LOCAL SPECTRAL THEORY FOR POSITIVE OPERATORS

This implies that

Ilx'IIA = sup{ {x', z) : z E ball(F('x))+}.

On the other hand, for all j E N it follows that

iiTjIF(A)ii = sup { (x', Tjz) : x' E ball(J('x, E'))+, z E ball(F('x))+}

= sup{(T'jx',z): x' E ball(J('x,E'))+,z E baU(F('x))+}

= \\T'jiJ(A,E')\\'

Hence r(TF(A») = r(T~(A,E'»)' We will show that r(TF(A») = 'x.

99

If this fails to be true, then it follows from the previous results that the norms II·IIA and II· liE' are equivalent on J(,X, E'). Since G is norm determinating and G c J ('x, E'), it follows that

IlxilE = sup{l(x',x)1 : x' E ball(E') n J(A, E')}.

Therefore, the norms II . 11:\ and II . liE are equivalent on E. This implies that F('x) = E which contradicts r(TF(A») < ,x :S r(TE)' It follows that r(TF(A») = ,x.

Furthermore, the construction of F('x) implies that the mapping ,x ----+ F()") is decreasing. Since r(TF(A») = ,x, this mapping is strictly decreasing on ]a,;3].

It remains to show that [a,;3] C (J(TE)' If this fails to be true, then there is ,x E (a,,6[, ,x rj. dTE)' By [3,4.1.1], R('x,TF) is positive, but R('x,TE) fails to be positive. It follows that there is x E E+ such that R(,X, TE)X rj. E+, but R(,X, TF)X E F+. This contradicts E+ = En F+. Q.E.D.

REMARK 1.14 The existence of some norm determinating ideal G C E~ does not imply that the norm of E is order continuous. If E = £00, then G = £1 c ball(N) is norm determinating and contained in E~.

COROLLARY 1.15 Let E be a Banach lattice with order continuous norm, F be a Banach lattice containing E as an ideal. /jT E £(Fh such that TE C E and r(TE) = ,6 > a = r(TF), then [a,,6] C (J(TE)' Moreover, for all ).. E (a,,6] there exists a T-invariant Banach lattice F('x), containing E as a dense ideal, such that r(TF (>,») = A.

Proof Let Q: E ----+ F be the embedding. By Nakano's theorem [3, 1.4.11, and by 1.4.19], G = Q' F' is an order dense ideal in E'. In particular, G is norm determinating with respect to E. The assertion follows from the preceding theorem. Q.E.D.

COROLLARY 1.16 Suppose that E is a Banach lattice with order continuous norm and T E £(E)+. Let F be a Banach lattice, contained in E as a dense


ideal, which is T-invariant and satisfies a = r(TF) < (3 = r(TE)' For all ). E (a, (3] there is a T'-invariant Banach lattice F().), containing E' as a dense ideal, such that r(T~(),)) = )..

Moreover, the mapping). --+ F()') is strictly decreasing on la, (3]. Proof It follows from [3,2.4.2] that E is an ideal in (E')~. Hence, G = F is

a norm-determinating ideal for E'. The result follows from Theorem 1.13. Q.E.D.

COROLLARY 1.17 Let E be a Banach lattice with order continuous norm and T E £(E)+. Suppose that F is a Banach lattice, containing En F as a dense ideal such that TF C F and a = r(TF) < (3 = r(TE)' For all A E (a, (3] there exists a Banach lattice H()') such that En F c H().) c E + F and r(TH(),)) = )..

Proof Set G = E + F. Fix). E (a, (3]. If ). ::; v(Ta), then Theorem 1.12 with G instead E and H().) = J().) shows that F is dense in G.

If veTa) < )., then Theorem 1.13, with G instead F and H().) = F().) , shows the assertion. Q.E.D.

References

1. I. Colojoaro, C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968.

2. K.H. Forster, B. \Nagy, On the local spectral theory for positive operators, Operator Theory, Advances and Applications 2 (1988), pp.71-81.

3. P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin-Heidelberg-New York, 1991. 4. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin

Heidelberg-New York, 1974. 5. A.C. Zaanen, Riesz spaces II, North-Holland, Amsterdam, 1984.


A Wiener-Young Type Theorem for Dual Semigroups

BEN DE PAGTER

101

Delft University of Technology, Faculty of Technical Mathematics and Informatics, Department of Pure Mathematics, P.O. Box 5031, 2600 GA Delft, The Netherlands

(Received: 27 Apri11992)

Abstract. The purpose of this paper is to obtain extensions of the Wiener-Young theorem for strongly continuous semigroups of positive operators in Banach lattices.

Mathematics Subject Classifications (1991): 46B42, 47D03

Key words: operator semigroup, Wiener-Young theorem

Introduction

Given a complex bounded Borel measure J1 on R we define for t E R the translated measure J1t by J1t(B) = J1(B - t) for all Borel sets B in R. We denote by 11J111 the total variation norm of J1. It is a classical result of A. Plessner (1929) that IIJ1t - J111 -+ 0 as t -+ 0 if and only if J1 is absolutely continuous with respect to the Lebesgue measure. This result was extended by N. Wiener and R.C. Young (1935), who showed that lim sup IIJ1t - J111 = 211J1811, where J18 is the

t-.o singular component of J1 with respect to the Lebesgue measure.

The above results can also be formulated within the framework of semi groups of linear operators. For t E R we define the translation operator T(t) in E =

Co(R) by T(t)f(x) = f(x + t) for x E R, where Co(R) denotes the space of all (complex valued) continuous functions on R vanishing at infinity. Then T*(t)J1 = J1t. where T*(t) denotes the adjoint of T(t) in Co(R)* = Mb(R), the space of all complex bounded Borel measures on R. The result of Plessner can now be reformulated as follows. Let EO be the domain of strong continuity of {T*(t)}tER' i.e.,

ast-+O}.

Then EO = Ma(R), the subspace of all measures which are absolutely continuous with respect to Lebesgue measure. Using the terminology of vector lattices,

102 BEN DE PAGTER

we can say that EO is a band in the dual E* = Mb(R) and we have a decomposition

E* = EO EB (E0)d,

where (E0)d = Ms(R), the subspace consisting of all measures which are singular with respect to Lebesgue measure. This decomposition is the Lebesgue decomposition. Moreover, the result of Wiener and Young can now be stated as lim sup IIT*(t).u - .ull = 211.usll, where.us is the component of.u in (E0)d.

t--.O The purpose of the present paper is to obtain extensions of the above results

for strongly continuous semigroups of positive operators in Banach lattices.

1. The General Framework

In this section we briefly recall the duality theory for strongly continuous semigroups of operators in Banach space. For details we refer to [2] or [4].

Let X be a (complex) Banach space. A collection {T(t) : 0 ::; t E R} =

{T(t) h>o of bounded linear operators in X is called a Co-semigroup if i) T(s + t) = T(s)T(t) for all s, t ?:: 0;

ii) T(O) = J; iii) T(t)x ---+ x in norm as t 10 for all x EX. Condition iii) is equivalent to saying that the mapping t f--+ T( t) is strongly continuous.

The generator A of a Co-semigroup {T(t) h::::o is defined by

A 1· T(t)x-x x = 1m -----'---'-------t!O t

for all x E dom A, where dom A is the set of those x E X for which this limit exists. Then dom A is a norm dense linear subspace of X and A is a closed linear operator. The type of {T( t) h;::::o is defined by

Wo = lim log IIT(t)ll. t->oo t

All .\ E C with Re.\ > Wo belong to the resolvent set p( A) of A, and for such .\ the resolvent of A is given by

R(.\,A)x = 100 e-AtT(t)xdt for all x E X.

Moreover .\R(.\, A)x ---+ x as .\ ---+ ex) (.\ E R). By X* we denote the (Banach) dual of X. For x* E X* and x E X we

denote by (x, x*) the action of x* on x. Given a Co-semigroup {T(t) h;::::o in X, the adjoint operators {T*(t) h;::::o form a semigroup of operators (Le., satisfy i) and ii) above). However, this semigroup is not, in general, strongly continuous (Le., does not satisfy condition iii) above). We note the following elementary facts:

WIENER-YOUNG TYPE TIIEOREM 103

a) {T*(t)k::o is w*-continuous, i.e., for each x* E X* the mapping t f---t

T*(t)x* is w*-continuous; b) The adjoint A * is closed, dom A * is w* -dense and A * is the w* -generator

of {T*(t)h::::o, Le.,

{ T*(t)x* - x* } dom A* = x* E X* : w* -lim exists

tlO t

A* * * l' T*(t)x* - x* * A* x = w - 1m for all x E dom ; tlO t

c) p(A *) = p(A) and R(A, A *) = R(A, A)* for all A E p(A). Although in general {T*(t) h::::o is not strongly continuous, we can always

consider the subspace of X* on which this semi group is strongly continuous. Let

X 8 = {x* E X* : IIT*(t)x* - x*11 -> 0 as t 1 O}

(this subspace is sometimes called the 'sun-dual' of X with respect to {T( t) h::::o). This space was introduced and investigated by R.S. Phillips ([6]).

Observe that: 1) X8 is a closed linear subspace of X*; 2) T*(t)(X8 ) C X8 for all t ~ 0; 3) If we define T8(t) = T*(t)lx0, then {T8(t)h::::o is a Co-semigroup in X8.

In the next Theorem we collect some basic properties of X8.

THEOREM 1.1 (see e.g. [4, Chapt. 14] or [2, Sect. 1.4.1]) If{T(t)h::::oisaCo-semigroup in Banach space X, then

i) X8 = domA* (norm closure); in particular, X8 is w*-dense in X*; ii) if we define

Ilxlh = sup{l(x,x*)1 : x* E X8, Ilx*11 ::; I}, then 11·111 is a norm in X, and there exists a constant M ~ 0 such that Ilx III ::; Ilxll ::; Mllxlh for all x E X (where we can take M = liminf IIAR()., A) II);

)'-too

iii) If we denote by A8 the generator of {T8(t)h>o, then domA8 = {x* E domA* : A*x* E X 8 },-

and A8x* = A*x* for all x* E domA8; iv) p(A8) = p(A*) = p(A) and R()", A8) = R().., A)*lx0 for all A E p(A).

Note that it follows from i) above that X8 = X* if and only if dom A * is norm dense in X*. In particular, if X is reflexive, then X8 = X*.

Next we give some simple examples to illustrate the above concepts and to motivate the results in which we are interested in the present paper.

EXAMPLE 1.2 A) 1. Let X = Co(R) be the space of all continuous functions on R vanishing at

infinity, with the sup-norm. For t E R the operator T(t) in Co(R) is defined

104 BEN DE PAGTER

by T(t)I(x) = f(x + t) for all x E R. Then {T(t)}tER is a Co-group in Co(R), whose generator is given by Af = I' with domain

dom A = {f E Co(R) : f is differentiable, f' E Co(R)}.

The dual space of Co(R) will be identified with the space Mb(R) of all bounded complex Borel measures on R, with total variation norm. Then T*(t)J1 = J1t. where J1t(B) = J1(B - t) for all Borel sets B in R. Denoting by 6x the point measure at x E R, we have T*(t)6x = 6xH, so that IIT*(t)6x - 6x ll = 2 for all t I- O. This shows that {T*(t)}tER is not strongly continuous on Mb(R). It follows easily that

domA* = {J1 E Mb(R) : DJ1 E Mb(R)},

and A*J1 = -DJ1 for all J1 E domA* (where DJ1 denotes the derivative of J1 in the sense of distributions). Using that a measure J1 E Mb(R) with DJ1 E Mb(R) is absolutely continuous with respect to Lebesgue measure m, and identifying these absolutely continuous measures, via the RadonNikodym theorem, with Ll (R, m), we get

dom A * = {g ELI (R, m) : 9 is of bounded variation}.

From this it follows, via Theorem l.li) that

X 8 = domA* = L1(R, m).

Note that T8(t)g(x) = g(x - t) m-a.e. on R. These results can also be formulated as follows. Let Ma(R) = {J1 E Mb(R): J1« m}, then

Ma(R) = X 8 = {J1 E Mb(R): IIJ1t - J111 ---+ 0 as t ---+ O},

and this is the classical result of A. Plessner ([7]). We denote by MAR) the subspace of all measures which are singular with respect to Lebesgue measure, i.e.,

Ms(R) = {J1 E Mb(R) : J1--Lm}. By the Lebesgue decomposition, any J1 E Mb(R) can be written uniquely as J1 = J1a + J1s, with J1a E Ma(R) and J1s E Ms(R), so Mb(R) = Ma(R) EB Ms(R). It was shown by N. Wiener and RC. Young ([9]) that

lim sup IIJ1t - J111 = 211J1s II t-+O

for all J1 E Mb(R), which is of course an extension of Plessner's result. In Section 3 of the present paper we will obtain a version of this result for Co-sernigroups of positive operators in certain Banach lattices.

B) Let X = Ll(T,m), where m is Lebesgue measure, and T will be identified with [0, 27r). Define T(t)f(O) = f(O+t) a.e. for all I E L1(T,m) and all t E R. Again {T(t)}tER is a Co-group. The dual of Ll (T, m) is, as usual, identified with Loo(T, m), and the adjoint group is given by T*(t)g(O) =

g(O - t) a.e .. Clearly, {T*(t)}tER is not strongly continuous, and it follows easily that X8 = C (T) in this case.

WIENER-YOUNG TYPE THEOREM 105

C) Let X = Co(R) as in Example A. We define {T(t) k:o in Co(R) by T(O) = I and T(t)f = Pt * f for t > 0, where

p. _! t t(Y) - 7f t2 + y2

is the Poisson kernel. Then {T( t) h;::o is a Co-semigroup, and the adjoint is given by T*(t)J.l = Pt * J.l for all J.l E Mb(R). One easily verifies that X8 = Ma(R). Note that T*(t)(X*) c X8 for all t > O.

2. Adjoints of Positive Co-Semigroups

All semigroups considered in the above examples are positive sernigroups in Banach lattices. We will consider such sernigroups and their duality theory in more detail. For the terminology and theory of Banach lattices we refer to the books [1], [8], [10].

Let E be a (complex) Banach lattice and {T( t) h;::o a positive Co-sernigroup in E (Le., {T(t)h>o is a Co-sernigroup consisting of positive operators). One of the first questions one may ask is whether or not E8 is a sublattice of E*. If {T* (t) h;::o consists of lattice homomorphisms, then it is clear that E8 is a sub lattice of E*. In general, however, this is not true, as was shown by A. Grabosch and R. Nagel ([3]).

In the next theorem we show that if E* has order continuous norm, then E8 is always a band in E* (note that this condition is satisfied for spaces E ::-: Co(n) with n an arbitrary locally compact space).

THEOREM 2.1 Let E be a Banach lattice such that E* has order continuous norm, and let {T( t) h>o be a positive Co-semigroup in E.

Then E8 is a baniin E*, and hence E* = E8 EB (E0)d. Proof Since E8 is closed and E* has order continuous norm, all we have

to show is that E8 is an ideal in E*. It clearly suffices to show that the solid hull of domA* is contained in E8. Moreover, since IR(A, A*)'P1 :s; R(A, A*)I'P1 for all 'P E E* (A > wo) it is sufficient to prove that 0 :s; 'P :s; 'IjJ in E* with 'IjJ E E8, implies that 'P in E8.

To this end, let

A = {x* E E* : :3t E [0,1] such that 0 :s; x* ::; T*(t)'IjJ}.

Since 'IjJ E E8, the set {T* ( t) 'IjJ : 0 :s; t :s; 1} is compact, and the order continuity of the norm in E* now implies that A is relatively weakly compact (see, e.g., [1, Thm. 13.8]). Hence {T*(t)<p: 0::; t ::; I} is relatively weakly compact. Since T*(t)<p ---+ <p (weak*) as t 1 0, it follows that T*(t)<p ---+ <p (weakly) as t 1 o. As is well kown, this implies that T* (t)<p ---+ 'P (norm) as t 1 0, and so <p E E8. Q.E.D.

106 BEN DE PAmER

In general, (E0)d is not invariant for T*(t), as is shown by Example 1.2.C. If, however, all T*(t) are lattice homomorphisms (e.g., if {T(t) }tER is a positive group), then it is clear that (E0)d is invariant.

Furthermore we note that if E is Dedekind O'-complete, then (E0)d = {a} (see [5]). Therefore, if E is Dedekind O'-complete and E* has order continuous norm, then EO = E*. This is for instance the case if E = (Co).

PROPOSITION 2.2 Let {T(t) h>o be a positive Co-semigroup in the Banach lattice E. Take ° ::; tp E (E0)d and ° ::; u E E. Then

(u, tp A T*(t)tp) = ° m-a.e. on [0,00).

Proof Take A > max(O, wo). Then tp A R(A, A)*tp = 0, so

inf{ (w, tp) + (u - w, R(A, A)*tp) : 0::; w ::; u} = 0.

Take E > 0. Then there exists ° ::; w ::; u such that

and (u - w, R(A, A)*tp) ::; E.

Note that the function t 1-+ (x, tp A T*(t)tp) is upper-semicontinuous on [0,00), and hence a Borel function, for every ° ::; x E E. Now we have

100 e->'t(u, tp 1\ T*(t)tp) dt

= 100 e->'t(w, tp 1\ T*(t)tp)dt + 1000 e->'t(u - w, tp A T*(t)tp) dt

::; 100 e->'t(w, tp) dt + 100 e->'t(u - w, T*(t)tp) dt

1 1 = ),,(w, tp) + (u - w, R(A, A)*tp) ::; ()" + l)E.

This holds for all E > 0, and so

100 e-At(u,tp 1\ T*(t)tp) dt = 0,

from which it follows that (u, tp 1\ T*(t)tp) = ° m-a.e. on [0,00). Q.E.D.

The following Theorem is one of the main ingredients for the extension of the Wiener-Young Theorem, to be obtained in the next Section.

THEOREM 2.3 Suppose that {T(t) h~o is a positive Co-semigroup in the Banach lattice E, and that E has a quasi-interior point or E* has order continuous norm.

Then T*(t)tp~tp m-a.e. on [0,00) for all tp E (E0)d.

WIENER-YOUNG TYPE TIlEOREM 107

Proof First note that it is sufficient to prove the result for ° ::; r.p E (E8)d, since IT*(t)x*1 ::; T*(t)lx*1 for all x* E E*.

Let 0::; r.p E (E8)d be fixed. First assume that E has a quasi-interior point uo. Then (uo, r.p /\ T*(t)r.p) = ° implies that (x, r.p /\ T*(t)r.p) = ° for all ° ::; x E E, and so r.p /\ T*(t)r.p = 0. Hence in this case the theorem is a direct consequence of Proposition 2.2.

Now assume that E* has order continuous norm. By [10, Thm. 125.1], for n = 1,2, ... there exist ° ::; Un E E such that ((Ixl - un) +, r.p) < 1/ n for all x in the unit ball BE of E. By Proposition 2.2, for each n = 1,2, ... there is a Lebesgue null set Nn such that (un,T*(t)r.p/\r.p) = ° for all t E [0, oo)\Nn. Let N = U~=I Nn. Then m(N) = 0, and (un, T*(t)r.p/\r.p) = ° for all t E [0,00) \N and all n = 1,2, .. "

Take x E BE, then Ixl ::; (Ixl - un )+ + Un for each n = 1,2, ... , and so for t E [0,00) \ N we find that

l(x,r.p/\T*(t)r.p)I::; (Ixl,r.p/\T*(t)r.p)

::; ((Ixl - un)+, r.p /\ T* (t)r.p) + (un, r.p /\ T* (t)r.p)

+ 1 ::; ((Ixl - Un) ,r.p) < -. n

Since this holds for all n = 1,2, ... we conclude that (x, r.p /\ T*(t)'P) = ° for all x E BE, and so 'P /\ T*(t)'P = 0, for all t E [0,00) \ N. Q.E.D.

3. A Wiener-Young Type Theorem

In this section we assume that E is a Banach lattice such that E* has order continuous norm. Let {T(t) h>o be a positive Co-semigroup in E. From Theorem 2.1 we know that E8 is a 6~md in E*, and so

E* = E8 e (E8)d.

Hence, every r.p E E* has a unique decomposition r.p = r.pI + r.p2 with r.pI E E8 and r.p2 E (E8)d. Observe that

lim sup IIT*(t)r.p - r.pll = lim sup IIT*(t)r.p2 - r.p211· tlO tlO

By Theorem 2.3 we have T*(t)r.p2-1-r.p2 m-a.e. on [0,00), and so

IT*(t)'P2 - 'P21 = IT*(t)'P21 + 1'P21 m-a.e. on [0,00).

This implies that IIT*(t)'P2 - 'P211 ?:: 11'P211 m-a.e. on [0,00). We thus get the following proposition.

PROPOSITION 3.1 Let E be a Banach lattice such that E* has order continuous norm, and let {T( t) h20 be a positive Co-semigroup in E. Then

11r.p211 ::; lim sup IIT*(t)r.p - r.pll ::; KIIr.p211 tlO

108 BEN DE PAmER

for all SO E E*, where S02 is the component of SO in (E0)d (and K is a constant depending on the semigroup; we can take K = lim sup IIT(t) II).

no If E is an AM -space, the above estimates can be improved. Indeed, in that

situation E* is an AL-space, i.e., the norm in E* is additive on disjoint elements (which also implies that the norm in E* is order continuous). Using the same notation as above, we have

m-a.e. on [0, ()()).

Moreover, T*(t)S02 ----+ S02 (weak*) as t 1 ° implies that limsuPtlO IIT*(t)soll > II so211· Therefore

lim sup IIT*(t)S02 - so211 2: 211so211· tlO

If we assume in addition that limtlo IIT(t)11 = 1, a condition which is automatically satisfied if E is an AM -space with unit, we find that

lim sup IIT*(t)S02 - so211 = 211so211· tlO

This proves the following theorem.

THEOREM 3.2 Let {T(t)}t;:::o, be a positive Co-semigroup in an AM-space E such that limtlO IIT( t) II = 1. Then

lim sup IIT*(t)SO - soil = 211so211 tlO

for all SO E E*, where S02 is the component of SO in (E0)d.

Note that the above result is in particular valid for every positive contraction semigroup in a space Co(n), with n locally compact, and for every positive Co-semigroup in a space C(n), with n compact.

If we take in the above theorem E = Co(R) and for {T(t)}tER the translation group we get precisely the theorem of Wiener and Young.

References

1. C.D. Aliprantis, O. Burkinshaw, Positive Operators, Acad. Press, London, 1985. 2. P.L. Butzer, H. Berens, Semigroups of Operators and Approximation, Springer-Verlag,

Berlin-Heidelberg-New York, 1967. 3. A. Grabosch, R. Nagel, Order structure of the semigroup dual: a counter example, lndag.

Math. 92 (1989), pp. 199-201. 4. E. Hille, R.S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq.

PubL 31, AMS, Providence, 1957. 5. J.M.A.M. van Neerven, B. de Pagter, Certain seruigroups on Banach function spaces and

their adjoints, in: Semigroup Theory and Evolution Equations, Lecture Notes in Pure and Applied Mathematics 135, Marcel Dekker, New York-Basel, 1991.

WIENER-YOUNG TYPE TIIEOREM 109

6. R.S. Phillips, The adjoint semi-group, Pac. 1. Math. 5 (1955), pp. 269-283. 7. A. Plessner, Eine Kennzeichnung der totalstetigen Funktionen, 1. fUr Reine und Angew.

Math. 60 (1929), pp. 26-32. 8. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin-

Heidelberg-New York, 1974. 9. N. Wiener, R.C. Young, The total variation of g(x + h) - g(x), Trans. Amer. Math. Soc.

33 (1935), pp. 327-340. 10. A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam-New York-Oxford, 1983.


Kri vine's Theorem and the Indices of a Banach Lattice

ANTON R. SCHEP Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A


III

Abstract. In this paper we shall present an exposition of a fundamental result due to J.L. Krivine about the local structure of a Banach lattice. In [3] Krivine proved that €p (1 S; P S; (0) is finitely lattice representable in any infinite dimensional Banach lattice. At the end of the introduction of [3] it is then stated that a value of p for which this holds is given by, what we will call below, the upper index of the Banach lattice. He states that this follows from the methods of his paper and of the paper [5] of Maurey and Pisier. One can ask whether the theorem also holds for p equal to the lower index of the Banach lattice. At first glance this is not obvious from [3], since many theorems in [3] have as a hypothesis that the upper index of the Banach lattice is finite. This can e.g. also be seen from the book [6] of H.D. Schwarz, where only the result for the upper index is stated, while both indices are discussed. One purpose of this paper is clarify this point and to present an exposition of all the ingredients of a proof of Krivine's theorem for both the upper and lower index of a Banach lattice. We first gather some definitions and state some properties of the indices of a Banach lattice. For a discussion of these indices we refer to the book of Zaanen[7].


Key words: Krivine's theorem, Banach lattice

1. Introduction

DEFINITION 1.1 Let 1 ::; p ::; 00. A Banach lattice E has the strong £pdecomposition property (or satisfies a lower p-estimate) if there exists a constant M such that for all disjoint elements Xl, ... ,Xn in E we have

for p < 00 and

n

max Ilxili ::; M LXi lS;tS;n i=l

in case p = 00.

112 ANTON R. SCHEP

Similarly we have

DEFINITION 1.2 Let 1 ::; p ::; 00. A Banach lattice E has the strong Rpcomposition property (or satisfies an upper p-estimate) if there exists a constant M such that for all disjoint elements Xl, ... , Xn in E we have

for p < 00 and

in case p = 00.

Obviously, any Banach lattice has the strong Roo-decomposition property and the strong Rl-composition property. By means of the above definitions we can define the (Grobler-Dodds) indices of a Banach lattice.

DEFINITION 1.3 Let E be a Banach lattice. Then the numbers

O"(E) = inf{p 2': 1 : E has the strong Rp-decomposition property}

and

s(E) = sup{p 2': 1 : E has the strong Rp-composition property}

are called the upper or lower index of E, respectively.

If dim(E) = 00, then 1 ::; s(E) ::; dE) ::; 00. We collect some basic facts about the indices of a Banach lattice. If dE) < 00, then E has order continuous norm and if s(E) > 1, then the dual space E* has order continuous norm. Also, we have

1 1 --+--=1 s(E) O"(E*)

and 1 1

--+--=1. O"(E) s(E*)

DEFINITION 1.4 Let E be a Banach lattice. Then E is said to contain a ...\lattice copy of Rp( n) if there exist disjoint elements Xl, ... ,xn in E such that for n-tuples {adi=l of real numbers we have

n

II{adllp::; Laixi ::; ...\11 {ai}llp· i=l

KRIVINE'S THEOREM AND INDICES OF A BANACH LATIICE 113

One easily sees that s(E) :::; p :::; cr(E) whenever E contains for all integers n a A-lattice copy of .ep(n), We now can state the result of JL Krivine.

THEOREM 1.5 (Krivine) Let E be an infinite dimensional Banach lattice. Then E contains for all integers n and all E > 0 an (1 + E)-lattice copy of .ep(n) for p = s(E) and p = cr(E).

In the next two sections we shall give an exposition of the main ingredients of the proof of this theorem.

2. The Maurey-Pisier reduction

In this section we shall show that the methods of Maurey-Pisier [5], adapted to disjoint sequences, allows one to reduce the theorem of Krivine to the case that E is a Banach lattice of sequences such that .eP- E C E c .eP+E with p = s(E) or p = cr(E). We begin with a more detailed study of disjoint sequences.

THEOREM 2.1 Let 1 :::; p :::; 00. Then either 1) There is an E > 0 such that for all integers n there are disjoint Xl, ... , Xn

in E such that for all n-tuples of real numbers n

maxlail:::; Laixi :::; (1 +E)II{adr=lllp; t<n - i=l

or 2) There are s < 00 and C > 0 such that for all disjoint Xl, ... ,Xn we have

(t, IIXill')" S Csup {lltaixl t lail' ~ I}

Moreover, if 1) holds, then 1) holds for all E > O.

Idea of the Proof Define an as

inf { a: l~~ Ilxill :::; a sup {lltaixill : III a;}II, ~ 1, Xl, ... , Xn disjoint in E} } .

Then it easy to show that an :::; 1 and that an is submultiplicative, i.e. amn :::; aman. Now either an = 1 for all n and case 1) holds for all E > 0, or there exists an integer no such that ano < 1 and case 2) holds for s > b, where

ano = n~(3. Hence either 1) holds for all E > 0 or 2) holds. Now note that if 1) holds for some E > 0, then 2) cannot hold, so that in this case 1) holds for all E > O. Q.E.D.

114 ANTON R. SCHEP

We note that if p = 00, then 2) says that E has the strong t's-decomposition property. Hence if cr(E) = 00, then for p = 00 we have case 1) in the above Theorem, i.e. Krivine's theorem for p = cr(E) = 00.

COROLLARY 2.2 The set of p E [1,00] for which 1) holds is a closed interval [l,po] for some Po E [1,00]

Note that if P < seE), then 1) holds for all normalized disjoint Xl, ... , Xn in E, while if P > cr(E), then 2) holds. Hence we have

s(E) ::; Po ::; cr(E).

In fact, we shall see that Po = cr( E). We now take a closer look at the constants involved in the inequality used to

define the t'p-decomposition property. Define for fixed n:

Cp,n = inf{C: (t IIXiIIP) ~ :::; C tXi , Xl,··· ,xn disjoint in E}. 1=1 1=1

It is easy to see that the sequence {Cp,n}~=l is submultiplicative. One can then 1 1

show, as in [5], that Cp,n 2': np- u(E) for p < cr(E), and that

lim 10gCp,n = ! __ 1_. n-->oo log n p cr( E)

We now relate the constant Cp,n to a norm inequality for positive linear maps from t'oo(n) -7 E.

PROPOSITION 2.3 Let p 2': 1 and T: t'oo(n) -7 E a positive linear operator. Then we have

for all Xl, ... , Xn E t'oo(n).

Now we can prove, as in [5], the following theorem, which can summarized by saying that Po = cr(E).

THEOREM 2.4 For p 2': 1 there are s < 00 and C > 0 such that for all disjoint Xl,·.·, Xn we have

if and only if p > cr( E).

KRIVINE'S TIIEOREM AND INDICES OF A BANACH LATTICE 115

In particular, we note that when p = cr( E), then by the above Theorem we have case 1) in Theorem 2.1. The proof of the above Theorem uses a factorization theorem ([5, Lemma O.1(a)]) through the Lorentz space .cp ,l(n) for operators from .coo(n) -+ E.

COROLLARY 2.5 1) Let p = (J(E). Then for all E > 0 and all n 2': 1 there are disjoint xf, ... ,x~ E E such that

n

!p.<axlail:::; Laixf :::; (1 +E)II{adf=lllp t n - i=l

for all n-tuples {ai} of real numbers.

2) Let p = s (E). Then for all E > 0 and all n 2': 1 there are disjoint yf, ... ,y~ E E such that

n

II{ad~lllp:::; Laiyf :::; (1 + E)II{adf=llh i=l

for all n-tuples {ai} of real numbers.

The proof of 1) of this Corollary follows from the above Theorem in conjunction with Theorem 2.1. Part 2) follows then via a duality argument. We note a consequence of the above Corollary, which is not used further on, but which gives another (new) characterization of the indices of a Banach lattice. As usual we shall denote by pi the conjugate exponent of p.

COROLLARY 2.6 Let E be a Banach lattice with indices s(E) and (J(E). Then the following hold.

1

1) p < s(E) if and only if n -p II Vk=l IXkll1 -+ 0 for all nann bounded se-quences {Xk}.

1

2) p > (J(E) if and only if L-:::=l n -illlxnli < 00 for all positive summable sequences {xn }.

Proof We start with proof of 1). Assume first p < s(E). Then let r be such that p < r < s(E). Then for any sequence {xd bounded in norm by 1 we have

as N -+ O. Hence for given E > 0 there is an N such that

116

for all M > N. Now

n-i tYI1Xklii ~ n-~ I[YIIIXklll + n-i tYN 1Xklll

"n-~ I :~: IXkl1 ' I k~1 dlxkl <2,

ANTON R. SCHEP

for n large enough. This proves the implication from left to right in O. Assume now that p 2 seE). By induction we shall construct a bounded

sequence {xd and a sequence of integers Nm i (X) such that

For m = 1 we take Xl E E with IlxI11 = land NI = 1. Assume now that NI < ... < N m and Xl, ... , X N m such that Xi ~ 1 have been chosen such that

J 2 -- J (*) holds. Then choose Nm+l > Nm such that 3Nm+I(Nm+1 - NmF 2: ~. Then by the above Corollary we can find disjoint Xk with Nm < k :::; Nm+l such that

Hence these x~s are also bounded in norm by 1 and putting all a~s = 1 we get

N rn+! 2 ! V IXkl 2 3(Nm+1 - Nm)"p. Nrn+l

Hence we have

Le. (*) holds for m + 1. This completes the proof of l). For the proof of 2) we first assume that p > O"(E). Taking O"(E) < r < p we

have

Assume now p s: O"(E). Then we can find integers Nk < Nk+l such that

Nk+! 1 sup L - = 00.

k n=Nk+1 n

KRIVINE'S THEOREM AND INDICES OF A BANACH LATTICE 117

By the above Corollary we can then find for each k disjoint XNk+1,···, XNk+l

such that

Let now {an} satisfy

N k +1

L anXn n=Nk+1

Then the sequence {Ianxnl} is summable, so we get by hypothesis that

00 1 00 1

L n -p7 lanl :=:; L n -p7 Ilanxnll < 00

n=l n=l

1

for all sequences {an} satisfying (**). It follows that {n - p7 } is in the dual space of the Banach space of sequences defined by (**). Hence we have

which contradicts the choice of the N£s. Q.E.D.

QUESTION 1 Does the above Corollary remain true if one restricts oneself to disjoint sequences?

If E and F are Banach lattices, then we say that F is finitely lattice representable in E if for all E > 0, all n and all disjoint YI,' .. ,Yn E F there are disjoint Xl, ... ,Xn in E such that

n

< ~a·x· - ~ ~ ~

F 1

n

:=:;(l+E) LaiYi ElF

We shall use the notation F '--J. E to denote the above relation between F and E. There is a standard way to get Coo '--J. E, as can be seen from the following proposition, the proof of which we leave to the reader. We denote by ei the sequence {O, ... , 1, 0, ... }, where we have a one on the ith position and zeros elsewhere.

PROPOSITION 2.7 Let {xl, ... ,x~} be a disjoint sequence in E such that: 1) Ilxill :=:; 2 for all i and n;

2) II2::f=l aixill 2: 6SUPlS:;iS:;n lail for some 6> ° and all (ai) E Coo·

118 ANTON R. SCHEP

Let II Li aiei II' = limnEU II Li aixfllE, where U is a free ultrafilter on E. Then 11·\\' defines a lattice norm on Coo. If F denotes the completion of Coo with respect to this norm, then F c .eoo and F <.......t E. Moreover, if F contains an (1 + E)-COPY of .ep(n) for all E > 0, then also E contains an (1 + E)-COPY of .ep(n) for all E > 0.

We apply the above construction to the disjoint sequences {xl" .. ,x~}, respectively {yr, ... ,y~}, of Corollary 2.5 with E = 1, to get two lattice norms 11·110" II· lis on Coo, respectively. We denote their completions by FO', Fs , respectively. From the above Proposition we get that Fs '-t E and FO' '-t E. Moreover,

Corollary 2.5 combined with the definitions of s(E) and dE) implies that if we let p = s(E) and q = dE), then for all E > ° there are constants C1 and C2

such that

and

I L aiei I ::; I L aiei ::; C21L aiei , , P , S , P-E

i.e. we have reduced Krivine's Theorem to the case that F is a Banach lattice of sequences such that .eP- E C F c .eP+E with p = s(E) or p = O'(E). We conclude this section by applying a theorem of Brunel-Sucheston [1], which shows that we can further improve on the norm of F.

THEOREM 2.8 (Brunel-Sucheston) Let X be a Banach space and Xn E X a bounded sequence which is not Cauchy. Then there is a subsequence {X7/-i} of {xn} such that

exists for all {an} E Coo and defines a norm on Coo.

REMARK 2.9 The proof of the above Theorem follows from an application of the infinite Ramsey Theorem. The norm obtained from the above Theorem is 'spreading' or 'subsymmetric', i.e. \I{ aI, a2, ... } II = II {O, ... ,aI, 0, ... ,0,

a2," ·}II· Applying the above construction to the Banach lattice of sequences F such

that .eP- E C F c .eP+E with p = s(E) or p = O'(E), we get a new Banach lattice of sequences which has all the properties of F and the additional property that the norm on F is spreading as indicated above. This final step completes the Maurey-Pisier reduction.

KRIVINE'S THEOREM AND INDICES OF A BANACH LATIICE 119

3. Krivine's Theorem

In this section we shall assume that F is a Banach lattice of sequences which satisfies all the properties obtained at the conclusion of the previous section. We shall show that fp is finitely lattice representable in F. Since F is finitely lattice representable in E, the Theorem of Krivine will follow. In our approach we will follow Lemberg's proof [4] of Krivine's more general theorem, and show that in the lattice context it produces the required lattice copy. We start by showing that we can represent F as a Banach lattice of sequences on the positive rationals instead of the positive integers. Let {qk} be a fixed enumeration of the positive rationals Q+. Then every y E coo(Q+) has a unique representation y = I:k=l akeqk' where eqk denotes the characteristic function of {qk} and where q1 < ... < qn· Define the norm of such an y by II I:k=l akekllF and let G denote the norm completion of Coo(Q+) with respect to this norm. It is a consequence of the spreading property that F and G are isometrically isomorphic as Banach lattices. Let G1 denote the closed subspace generated by {eq : q E Q+ n [0, I]}. We define the linear operators T and U on G 1 by T(eq ) = e~ + e(q+!) and

2

U(eq ) = e9..3 + eq+! + eq+2. One easily verifies the following proposition. 3 3

PROPOSITION 3.1 Let G 1, T and U be as above. Then the following properties hold.

1) TU = UT; 2) 11/11::; IITIII ::; 211/11 for all I E G1;

3) 11I11 ::; IIU III ::; 311/11 for all I E G 1; ! !

4) reT) = 21' and r(U) = 31'.

!

We note that since T 2:: 0, we get by 4) that 21' E (J(T). For a proof of the following Proposition we refer to [4].

PROPOSITION 3.2 Let X be a Banach space and let T and U be commuting bounded linear operators on X. Let A EdT) with IAI = reT). Then there are Xn E X with IIxnll = 1 and M E (J(E) such that IITxn - AXnll --+ 0 and IIUxn - MXnl1 --+ ° as n --+ 00.

In our application of the above Proposition we take X = G1, T, U as above

and A = 2i to get: There are In = I:~n Q:keqk in G1 with IIlnllc! = 1 such that !

liT In - 21' In II --+ 0 and IIU In - MIn II --+ 0 as n --+ 00. By replacing In by Ilnl we can assume that In 2:: 0 and that M 2:: O. Define now b~ = I:~n lakleqk+m. Then b~ E G has norm one and for fixed n the sequence {b~} is disjoint. We now define one more norm on Coo, as follows: for I: aiei E Coo let

120 ANTON R. SCHEP

where U is a free ultrafilter on N. As before it is easy to verify that 11'11* defines a lattice norm on Coo with the property that if H denotes the completion of Coo with respect to 11'11*, then H "---+ C. The proof of Krivine's Theorem is now completed exactly as in [4] (except that the A of [4] is known in advance to be

1

21'). One obtains as in [4, Prop. 11.7] the following Proposition.

PROPOSITION 3.3 The norm 11'11* = II· Ib on H, i.e. H = 1!p.

From Krivine's Theorem we know that if p = seE) or p = (J(E), then E contains a (1 + E)-lattice copy of 1!p(n) for all E > 0 and all n 2: 1. As observed before, E can contain A-lattice copies of fp(n) for all, only if s(E) ~ p ~ (J(E). If seE) < (J(E), one can therefore ask whether there are always values of p, other than the end points seE) and (J(E), for which the conclusion of Krivine's Theorem hold. We show by means of an example that that is not the case in general.

EXAMPLE 3.4 Let E = Lp(O, (0) n Lq(O, (0) with 1 :::; p < q :::; 00 and Ilxll = max{llxllp, Ilxll q}· It is easy to verify that in this case seE) = p and (J(E) = q. Assume that p < r < q exists such that E contains a (1 + E)-COPY of 1!r (n) for all E > 0 and all n 2': l. Then for all n E N we can find disjoint Xl,n,'" ,xn,n in E with max{llxi,nllp, Ilxi,nllq} = 1 such that

II {Ui} lie" max { (t,IUi I' IIXi,n IIP)* , (t,ladq Ilxi,n Ilq ) i }

:::; 211{adllr

(1)

for all m 2: 1 and all {ad E Rm. Let U be a free ultrafilter on N and define Ai = limnEu Ilxi,nllp and l1i = limnEu Ilxi,nllq· Then we have max{Ai, I1d = 1 for all i and by (1),

for all m and all {ad E Rm. In particular, we have

1

(~laiIPIIAfIIP) l' :::; 211{adllr (3)

---1L

for all m and all {ad E Rm. Now take ai = A;-P in (3) to obtain

KRIVINE'S THEOREM AND INDICES OF A BANACH LATTICE 121

which implies that Ai --+ 0 as i --+ 00. Hence there is io such that Jti = 1 for all i .2: io. By restricting ourselves to i .2: io in (2) we can assume that Jti = 1 for all i in (2). Then we take ai = * and ai+l = 1 - * in (2) to find that for all n> 1 we have

This implies that Ai+ 1 .2: 1 for all i, which contradicts Ai --+ O.

REMARK 3.5 The methods presented above this example can be easily modified to show that one can also take E = fp EEl f q, where 1 S p < q S 00 with II(x, y)11 = max{llxllp1 Ilxll q }· We also note that a special case of the above example, proved by a completely different argument, was given in [2, p. 288].

Acknowledgment

The author of this paper wishes to express his gratitude to Peter G. Dodds for the many stimulating discussions he had during the preparatory work for this paper, while the author was a Visiting Research Fellow at the Flinders University.

References

1. A. BruneI, L. Sucheston, On B-convex Banach spaces, Math. Systems Th. 7 (1973), pp.294-299

2. W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 217 (1979), pp. 1-298

3. J.L. Krivine, Sous-espaces de dimension finie des espaces de Banach n~ticules, Ann. of Math. 104 (1976), pp. 1-29

4. H. Lemberg, Nouvelle demonstration d'un tbeoreme de J.L. Krivine sur la finie representation de Rp dans un espace de Banach, Isr. 1. of Math. 39 (1981), pp. 341-348

5. B. Maurey, G. Pisier, Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach, Studia Math. 58 (1976), pp. 45-90

6. Hans-Ulrich Schwarz, Banach Lattices and Operators, Teubner-Texte zur Mathematik, Band 71, B.G. Teubner, Leibzig, 1984.

7. A.C. Zaanen, Riesz spaces II, North-Holland, Amsterdam-New York-Oxford, 1983.


Representations of Archimedean Riesz Spaces by Continuous Functions

A.W. WICKSTEAD

123

Department of Pure Mathematics, The Queen's University of Belfast, Belfast BT7 iNN, N.I., u.K.


Abstract. A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces


Key words: representation, Archimedean Riesz space

1. Introduction

The first examples of Archimedean Riesz spaces that one meets are all spaces of real-valued functions on some set. Examples are the space of continuous realvalued functions on a compact Hausdorff space, the various sequence spaces including not only the Banach spaces Co, c, foo and fp but also <1>, the space of all sequences with only finitely many non-zero terms, and s, the space of all real sequences. Slightly later one meets the Banach spaces Loo(j.t), of all p,essentially bounded functions, V(p,), of p-th power integrable functions, and the space M(p,) of all equivalence classes of JL-measurable functions which cannot be given a norm compatible with its order structure. It does not take one long to realise that some of these last spaces are, in some ways, very different from the earlier examples of Archimedean Riesz spaces. For example point evaluation functionals are Riesz homomorphisms into the reals on any space of real-valued functions, whilst there is certainly no Riesz homomorphism of M(p,) into the reals.

Since spaces of real-valued functions seem to be relatively easy to handle, as well as being a nice concrete object to try to understand, the existence of these rather unfriendly Archimedean Riesz spaces seems rather a pity. However, all is not lost! If we allow ourselves to consider functions taking values in a slightly more general object than the reals then it is possible to identify even M (p,) with a space of functions. Indeed this is possible for all Archimedean Riesz spaces. There are many proofs of this result which each give slightly

124 A.w. WICKSTEAD

different representations. I will not be concerned with the technicalities of how these results are proved or exactly what the differences between them all are-I will solely be concerned with how much use they are in proving results about Archimedean Riesz spaces.

This article is not meant to be a complete survey of the topic of representations of Archimedean Riesz spaces by continuous functions, it is merely a presentation of those that I have found useful, which I think might be useful to others or which are of historical interest. There are a number of results presented since, as one specialises the Archimedean Riesz space, the representation theorems say more and are thus more useful. I have also omitted discussion of representations of Archimedean Riesz spaces in spaces of measurable functions or in spaces of measures.

Even if you are amongst those who refuse for ideological reasons to use representations in the course of your proofs, these representations are still invaluable in deciding what results are likely to be true-a good starting point in any mathematical investigation.

The extended real numbers, Roo, comprise the reals and the two symbols 00 and -00 with arithmetic extended in the obvious manner and ordered by -00 < x < 00 for all x E R. It is usual to define ° x ±oo = ±oo x ° = 0, but there are occasions when it is preferable to say that such products are undefined. Usually this difference won't matter. ROO is topologised by the order topology which has as a subbase for the open sets all intervals of the form {x E Roo: x> a} or {x E ROO: x < a}. This makes the map n:Roo -+ [-1,1]' defined by n(x) = H::[X[ into an order preserving homeomorphism.

If X is a topological space then coo(X) denotes the set of all continuous ROO-valued functions, f, on X such that f-l(R) is dense in X. The pointwise partial order, (f 2: 9 {:} f (x) 2: 9 (x) for all x E X) makes Coo (X) into a lattice. Coo(X) is closed under multiplication by reals. The obvious way to try to define an addition on Coo (X) is to ask that (f + g) (x) = f (x) + g( x) whenever both f(x) and g(x) are real, but this is not always possible. However there may be many subsets of Coo(X) which are Archimedean Riesz spaces under these operations.

If one assumes that the topological space X is extremally disconnected, i.e. the closure of every open set is open, then it can be shown that any continuous ROO-valued function defined on a dense open subset of X may be extended continuously to the whole of X. It is then easy to see that COO (X) will itself be an Archimedean Riesz space.

More details on COO (X) may be found in [21, §§44, 47]. The results were originally proved in [35], [36] and [37].

By a representation of an Archimedean Riesz space, E, we will mean a subset it of some space coo(X) which is a linear space and a sublattice (for some topological space X) together with a linear lattice isomorphism f 1-+ j of

REPRESENTATIONS OF ARCHIMEDEAN RIESZ SPACES 125

E onto E. We will not, in general, make any assumption about infinite suprema and infima.

All Archimedean Riesz spaces have a representation in this sense. Much of the art of using representations is in picking the right one to use.

2. Archimedean Riesz Spaces

A strong order unit for a Riesz space, E, is an element e with the property that for any x E E there is a real number ,\ such that Ixl ::; '\e. The space Cb(X), of all bounded real-valued functions on a topological space X has a strong order unit, as do the sequence spaces C and £00' In each case a function is a strong order unit if and only if it is bounded away from 0 (for example the constantly one function). For Archimedean Riesz spaces with a strong order unit there is a very powerful and explicit representation theorem proved virtually simultaneously in [16], [18], [32] and [45]. There is a proof in [21, §45].

THEOREM 2.1 If E is an Archimedean Riesz space with a strong order uni! e, then there is a compact Hausdoif{ space, X, and a lattice isomorphism f f--t f of E onto a subspace of C(X) that is dense for the supremum norm. Furthermore we may assume that e is the constantly one function.

This is an extremely strong result for many reasons. The space X is uniquely defined (up to homeomorphism) and the functions in E separate the points of X. The density of E in C(X) can be improved to E = C(X) if E is complete in the norm

Ilflle = inf{'\: Ixl ::; '\e}.

The result only applies to a very special kind of Archimedean Riesz space, but every Archimedean Riesz space has many subspaces to which it can be applied, including any principal ideal, i.e. the smallest ideal, Ee, containing a positive element e E E. In Dedekind CT-complete Riesz spaces, or in Banach lattices, every principal ideal Ee is II· lie-complete so may be identified with some space C(X).

Among other simply proved consequences of this theorem is the existence of the Stone-Cech compactification of a completely regular space.

A weak order unit for an Archimedean Riesz space is a positive element e E E such that x 1- e '* x = O. Equivalently, the band generated bye, edd , is the whole of E. The sequence (lin) in Co is a weak order unit, but the space has no strong order unit. The space has no weak order unit. The classical representation for Archimedean Riesz spaces with a weak order unit is due to Maeda and Ogasawara [23] but this does not give rise to a small enough representation space to be of much practical use (see also the results of Vulikh in [40], [41] and [42]). However, by identifying points of the representation space that are not separated by points of ft, we obtain:

126 A.W. WICKS1EAD

THEOREM 2.2 If E is an Archimedean Riesz space with a weak order unit e, then there is a compact Hausdorff space, X, and a lattice isomorphism 1> of E onto an order dense subspace of c= (X) such that 1>( e) is the constantly one function and 1>(Ee) is dense in C(X) for the supremum norm.

In particular if the ideal Ee is II . lie-complete, then Ee is the whole of C(X), so that it is possible to apply the whole battery of techniques available for constructing elements of C(X) to finding elements of Ee. If, further, E is Dedekind iT-complete, then E is an ideal in C=(X). This result has been rediscovered many times over the years. One way to prove it is by extending the representation of Ee provided by Theorem 2.1. See Meyer [27] for this approach.

There are many results that give a representation of a general Archimedean Riesz space. The one that we present has some claim to being a canonical one and dates back, in this form, to the book by Kantorovich, Vulikh and Pinsker [17], although much of it is to be found in the work of Maeda and Ogasawara in [23] (see also Bemau [2]). Recall [21, §47] that a compact Hausdorff space, X, is extremally disconnected if and only if C=(X) is Dedekind complete.

THEOREM 2.3 Every Archimedean Riesz space, E, has a representation as an order dense sublattice, E, of C=(S) for some compact Hausdorff extremally disconnected space S. All suprema and infima are preserved by this identification and E is an ideal in C= (S) if and only if E is Dedekind complete.

Even though the space S in this theorem is unique to within homeomorphism, there are still unsatisfactory aspects of the Theorem. If you start with C([O, I]) this representation will essentially embed it into its Dedekind completion so will be a small subspace, whereas C([O, I]) was quite nice enough to start with. This kind of problem seems to be inevitable when one tries to work with a wide class of Riesz spaces. I have usually found it more profitable to work with suitable 'nice' ideals and then represent them. This is even more true when it comes to representations of Banach lattices.

The order dual, E~, of an Archimedean Riesz space, E, is the space of all linear functionals on E which are the difference of two positive linear functionals. It is the largest candidate for a dual of E, though smaller ones are also studied. It is desirable to be able to represent both E and E~ as functions on the same topological space in such a way that the action of a linear functional f E E~ on x E E has a natural interpretation. This is possible when E~ separates the points of E but I know of no more general version of this result (but see Vulikh's paper [43] which contains related results). This result was stated by the author in [44], but all the hard work of the proof is in [10], which contains a more general result worthy of inspection. Presumably it would also be possible to base a proof of this on Theorem 2.3.


THEOREM 2.4 Let E be an Archimedean Riesz space such that E~ separates the points of E. There is a locally compact, extremally disconnected Hausdorff space, S, and a Radon measure fJ, on S, for which nowhere dense sets are locally fJ,-negligible, such that there are representations f f-----+ j of E into C oo (8) and ¢ f-----+ ¢ of E~ onto an ideal in Coo (S) such that for each x E E and f E E~ we have

The author used this in [44] to study the duality theory of orthomorphisms.

3. Banach Lattices

Historically there have been two thrusts in the study of representations. The earliest was the axiomatic characterization of certain concrete Banach lattices, whilst more lately there have been attempts to find representations which will be of use in the study of more general Banach lattices.

Recall that a Banach lattice is a Banach space, E, which is also a Riesz space and such that, for f, gEE, If I ::; Igl '* Ilfll ::; Ilgll. The classical Banach spaces, C(X) and V(fJ,) for 1 :S p ::; 00 are all Banach lattices when given their usual norm and order. A Banach lattice with the additional property that for any two disjoint elements f and g, Ilf + gil = max{llfll, Ilgll}, is called an AMspace. The spaces C(X) with the supremum norm, and their closed sublattices, are examples of AM-spaces. The fact that these are, up to isomorphism, the only ones was proved virtually simultaneously by Kakutani and Bohnenblust in [6] and [16], M. and S. Krein in [18] and Nakano in [32] and [33]:

THEOREM 3.1 For any AM-space, E, there is a compact Hausdorff space, X, and an isometric Riesz isomorphism of E onto a closed sublattice of C(X).

A proof can be found in [46, Chap. 17]. The following theorem of Nakano in [33] deserves to be better known:

THEOREM 3.2 If E is an AM-space with the property that, for any set AcE which is bounded above,

sup{llall : a E A} = inf{llbll : b is an upper bound for A},

then there is a locally compact Hausdorff space, X, such that E is isometrically Riesz isomorphic to Co(X).

If you are really interested in AM-spaces I heartily recommend Goullet de Rugy's paper [13].

If 1 ::; p < 00, then an abstract V -space is a Banach lattice with the property that for any two disjoint elements f and g, Ilf + gilP = IlfilP + IlgiIP. Abstract

128 A.W. WICKSTEAD

Ll-spaces are more commonly called AL-spaces. It is certainly possible to give a representation of these spaces as spaces of continuous functions, but the following theorem is more useful:

THEOREM 3.3 If 1 :::; p < 00 and E is an abstract V-space, then E is isomorphic as a Riesz space to some space V(p,) for some Radon measure p,.

The case p = 1 was dealt with early on by Kakutani [15], whilst Bohnenblust [5] and Nakano [34] soon handled the other cases for separable spaces. A number of people contributed to the generalisation of this including Bernau [3], Bretagnolle, Dacunha-Castelle and Krivine [7], Gordon [11] and Marti [24]. The first complete published proof seems to be by Lacey and Bernau [19] and this is also proved in [46, Chap. 17]. This chapter also gives other results on the representation of Banach lattices by spaces of measurable functions.

One might look for other classes of Banach lattice defined by similar conditions expressing the norm of sums of disjoint elements in terms of their individual norms. Any such class will have no interesting members because of a theorem of Bohnenblust [5]:

THEOREM 3.4 Let E be a Dedekind cr-complete Banach lattice with a weak order unit and of dimension at least 3. If there is a function x(·, .) such that for any disjoint elements! and gin E, II! + gil = x(II!II, IIgll), then E is either an abstract V space for 1 :::; p < 00 or an AM-space.

A positive element in a Banach lattice, E, is called a topological order unit or a quasi-interior point of the positive cone if the norm-closed ideal generated by e is the whole of E, i.e. Ee = E. Every separable Banach lattice has a topological order unit and so have many others. Not all Banach lattices have topological order units, for example the continuous functions vanishing at infinity on a locally compact, non-cr-compact Hausdorff space. There is a very strong representation theorem for these spaces:

THEOREM 3.5 Let E be a Banach lattice with topological order unit e. There is a compact Hausdorff space X such that E is isomorphic as a Riesz space to a linear ideal in COO(X) with the principal ideal Ee being mapped onto C(X).

This Theorem is due independently to Davies [8] and [9], Lotz [20] and Goullet de Rugy [12] (and indeed follows easily from results in [42]). It has been extended by Schaefer [38] and an account of the construction is contained in [39]. It is possible to prove rather more than is stated here. The dual of E may be represented by a space of Radon measures on X and Nagel [30] has characterised order continuity of the norm in E in terms of this representation. His papers [29] and [31] may also be of interest in this context.

Again it is possible in many cases to use this representation for the study of general Banach lattices by restricting one's attention to principal closed ideals.


There seems to be no general representation theorem for all Banach lattices that is of more use than Theorems 2.3 and 2.4. There are, however, many results for special cases that I have not mentioned. Perhaps the most significant of these is the representation theorem for injective Banach lattices found in [14].

4. Band Preserving Operators on Banach Lattices

This final section is included as an example of using representations in proving non-trivial results about Archimedean Riesz spaces. The results proved are all known--{)Uf reason for including them is to show how representations may be used in real mathematics.

A linear operator, T, on a Riesz sp~e is called band preserving if it leaves every band invariant, i.e. if I ..1 g then T I ..1 g. An orthomorphism is an order bounded band preserving operator. It is equivalent that the operator is the difference between two positive band preserving operators. In fact, the orthomorphisms form a vector lattice with the property that if I ;:: 0, then T+ (f) = (T f) +. Orthomorphisms on function lattices may usually be represented as multiplication by some fixed function. For example on Co(X), for X a locally compact Hausdorff space, the orthomorphisms are multiplication by a bounded continuous function on X; on V (p) they are multiplication by elements of L (Xl (p) and on (J) they are multiplication by elements of s. There are examples known of band-preserving operators which are not orthomorphisms-probablY the simplest example is due to Meyer [28] (also presented in [4]). It, and proofs of the other results stated here, may be found in [46, Chap. 20].

Many of the classical examples of Riesz spaces have only order bounded band preserving operators defined on them, so it is of interest to try to find wide classes of Riesz spaces on which every band preserving operator is order bounded. One important class of Riesz space with this property is the class of Banach lattices. This was first proved by Abramovich, Veksler and Koldunov in [1]. It follows also from results in [26]. We present here an unpublished proof of this result due to the author. Similar ideas are involved in the proof of [22, Thm. 9.9], but here we emphasise how elementary the proofs are provided one is prepared to accept the groundwork that has been done over the last half a century in proving the representation theorems for us. The presentation of this proof is based on that in [25].

LEMMA 4.1 Let E be a uniformly complete Archimedean Riesz space and suppose that T is a non-order bounded band preserving linear operator on E. There is an order bounded disjoint sequence Un) in E with (TIn - n3 lInl) + > 0 for all n.

Proof. It suffices to consider the case that E has a weak order unit, for if the restriction to each principal band, edd , were order bounded then we could form 71~dd and piece these together to form T+. Let e be a weak order unit for E

130 A.W. WICKSTEAD

and represent E on a compact Hausdorff space X using Theorem 2.2. To make notation easier, we will assume that E is a linear sublattice of G<Xl(X) containing C(X).

Note first that if an element of E, J, vanishes on an open set U and x E U then, by Urysohn's lemma, there is a function g E

C (X) ~ E supported by U and with g(x) f O. As f vanishes on U we have J 1. g so that T J -.L g and hence T J(x) = O. Thus if J vanishes on an open set U then so does T J. Applying this argument to J - J' we see

U Z that if J and J' coincide on an open set U then so do T J and TJ' .

If it were true that T f (x) = 0 whenever f (x) = 0 then it would be not difficult to show that if f(x) f. 0 f. g(x) then Tf(x)/ f(x) = Tg(x)/g(x) so that T is just multiplication by some function ¢, where ¢( x) = T f (x) / J (x) for any f with f(x) f. O. Thus T would certainly be order bounded. Thus we may suppose that there is Xo E X and fEE with f(xo) = 0 but Tf(xo) > 1. Let U be the open set {x EX: T f (x) > I}. The point Xo must be in the closure of the set {x: f (x) > O} else J vanishes on a neighbourhood of Xo and hence so does TJ.

Tf f

We may thus define a disjoint sequence, (Un), of open subsets of U, with IJIlun < 1/n3 and Xo rt. Un, as follows. Choose Xl E U with IJ(xr) I < 1 and let UI be an open neighbourhood of Xl with IJllUl < 1 and Xo rt. UI .

If we have defined UI , ... , U n-l

let Xn be outside Uk:t Uk with IJI(xn) < 1/n3• Let Un be an open neighbourhood of Xn disjoint from each Uk for k = I, ... , n - 1; with Xo not in its closure and with IJIlun < 1/ n 3. Let us also define, for each n, Wn to be a non-empty open set with Wn ~ Un.


Tf f By Urysohn's lemma we may

construct <Pn E C (X) with

0"50 <Pn "50 1

<PnlWn == 1

supp <Pn ~ Un.

If fn = <Pn' f then the fn's are disjoint, Ifni "50 1/ n3 and fnlWn = flwn,

Consequently, we have

TfnlWn = Tflwn

> 11Wn

> n3lfnllwn,

I.e. T fn i n31fnl so that Un) is the desired sequence.

THEOREM 4.2 Every band preserving linear operator on a Banach lattice is order bounded.

Proof Let T be a non-order bounded band preserving linear operator on a Banach lattice E. Find a disjoint sequence fn in E with (T fn - n3 Ifnl)+ > 0 for all n. Let B be the band generated by the set {(Tfn - n3 Ifnl)-: n EN}. It is standard that the quotient E / B may be made into a Banach lattice and (since B is T -invariant) that T induces a band-preserving operator, T on E / B by the formula

T[fl = [Tfl,

where [Il denotes the equivalence class of f in E/ B. If fn were an element of B then Tfn - n31fnl would be in Bdd and hence (Tfn - n3 Ifnl)+ E

Bdd, contradicting the fact that (T fn - n3Ifnl)+ E Bd. Thus [Inl # O. As

132 A.W. WICKS1EAD

[Tin - n3 1inll = [(Tin - n3 Iinl)+l > 0 we have [Tinl > n3 [1inll. Let gn = [inl/n2 11 [inlll E E / B, so that (gn) is a disjoint sequence with Ilgnll = 1/n2

and IITgnl1 ?: n. Form 9 = L~l gk E E/B, which we may do as E/B is complete. Since 9 - gn .-l gn, T(g - gn) .-l Tgn, so that (Tg)+ 2: (Tgn)+ for each n EN. Since Tgn 2: n3gn > 0 we see that

for all n E N, which is a clear contradiction. Q.E.D.

References

1. Y.A. Abramovich, A.I. Veksler, A.V. Koldunov, On operators preserving disjointness, Soviet Math. Dokl. 20 (1979), pp. 1089-1093.

2. S.J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15 (1965), pp.599-631.

3. S.J. Bemau, A note on Lp-spaces, Math. Ann. 200 (1973), pp. 281-286. 4. S.J. Bemau, Orthomorphisms of Archimedean vector lattices, Proc. Camb. Phil. Soc. 89

(1979), pp. 119-128. 5. H.E Bohnenblust, On axiomatic characterisation of Lp spaces, Duke Math. 1. 6 (1940),

pp. 627-640. 6. H.P. Bohnenblust, S. Kakutani, Concrete representations of (M)-spaces, Ann. Math. 42

(1941), pp. 1025-1028. 7. 1. Bretagnolle, D. Dacunha-Castelle, J.L. Krivine, Lois stables et espaces LP, Ann. Inst.

H. Poincare 2 (1965/6), pp.231-259. 8. E.B. Davies, The structure and ideal theory of the predual of a Banach lattice, Trans.

Amer. Math. Soc. 131 (1968), pp. 544-555. 9. E.B. Davies, The Choquet theory and representation of ordered Banach spaces, Illinois 1.

Math. 13 (1969), pp. 176-187. 10. D.H. Fremlin, Abstract Kothe spaces II, Proc. Cam. Phil. Soc. 63 (1967), pp. 951-956. 11. H. Gordon, Measures defined by abstract Lp spaces, Pacific 1. Math. 10 (1960), pp.

557-562. 12. A. Goullet de Rugy, La theorie des cOnes bireticules, Ann. Inst. Fourier (Grenoble) 21

(1971), pp. 1-18. 13. A. Goullet de Rugy, La structure ideale des M-espaces, 1. Maths. Pures et Appl. 51

(1972), pp. 331-373. 14. R. Haydon, Injective Banach lattices, Math. Z. 156 (1977), pp. 19-47. 15. S. Kakutani, Concrete representation of abstract L-spaces and the mean ergodic theorem,

Ann. Math. 42 (1941), pp. 523-537. 16. S. Kakutani, Concrete representations of abstract M-spaces, Ann. Math. 42 (1941), pp.

994-1024. 17. L.v. Kantorovich, B.z. Vulikh, A.G. Pinsker, Functional Analysis in Partially Ordered

Spaces (Russian), Gostekhizdat, Moscow, 1950. 18. M. Krein, S. Krein, On an inner characterisation of the set of all continuous functions

defined on a bicompact Hausdorff space, C.R. Acad. Sci. URSS 27 (1940), pp.427-430. 19. H.E. Lacey, S.l. Bemau, Characterisations and classifications of some classical Banach

spaces, Adv. in Math. 12 (1974), pp. 367-401. 20. H.P. Lotz, Zur Idealstruktur in Banachverbanden, Habilitationsschrift Tubingen (1969). 21. W. A. 1. Luxemburg, A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam-London,

1971. 22. W.A.l. Luxemburg, Some Aspects of the Theory of Riesz Spaces, Univ. of Arkansas

Lecture Notes, Fayetteville (1979).


23. F. Maeda, T. Ogasawara, Representation of vector lattices, 1. Sci. Hiroshima Univ. 12 (1942), pp. 17-35.

24. J.T. Marti, Topological representations of abstract Lp-spaces, Math. Ann. 185 (1970), pp. 315-321.

25. P.T.N. McPolin, Disjointness preserving linear mappings on a vector lattice, Ph.D. Thesis, Q. U.B. (1983).

26. P.T.N. McPolin, A.W. Wickstead, The order boundedness of band preserving operators on uniformly complete vector lattices, Math. Proc. Cam. Phil. Soc. 97 (1985), pp. 481-487.

27. M. Meyer, Representations des espaces vectoriels reticules, Seminaire Choquet 13 (1973/4), pp. 1-12.

28. M. Meyer, Quelques proprietes des homomorphismes d'espaces vectoriels reticuh!s, E.R.A. Universite Paris VI 294 (1978).

29. R.J. Nagel, Darstellung von Verbandsoperatoren auf Banach-verbanden, Rev. Acad. Ci. Zaragoza 27 (1972), pp. 281-288.

30. RJ. Nagel, Ordnungstetigkeit in Banachverbanden, Manuscripta Math 9 (1973), pp. 9-27.

31. R.J. Nagel, A Stone-Weierstrass theorem for Banach lattices, Studia Math. 47 (1973), pp.75-82.

32. H. Nakano, Uber die Charakterisierung des allgemeinen C-Raumes, Proc. Imp. Acad. Tokyo 17 (1941), pp.301-307.

33. H. Nakano, Uber die Charakterisierung des allgemeinen C-Raumes II, Proc. Imp. Acad. Tokyo 18 (1942), pp. 280-286.

34. H. Nakano, Stetige lineare Funktionale auf dem teilweisgeordneten Modul, 1. Fac. Sci. Imp. Univ. Tokyo 4 (1942), pp. 201-382.

35. H. Nakano, Uber das System aller stetiger Funktionen auf ein topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), pp.308-31O.

36. T.Ogasawara, Theory of vector lattices I, 1. Sci. Hiroshima Univ. 12 (1942), pp.37-100. 37. T. Ogasawara, Theory of vector lattices II, 1. Sci. Hiroshima Univ. 13 (1944), pp.

41-161. 38. H.H. Schaefer, On the representation of Banach lattices by continuous numerical functions,

Math. Z. 125 (1972), pp. 215-232. 39. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin-

Heidelberg-New York, 1974. 40. B.z. Vulikh, Concrete representation of partially ordered linear spaces (Russian), Doklady

Akad. Nauk USSR 58 (1947), pp.733-736. 41. B.z. Vulikh, On concrete representation of partially ordered lineals (Russian), Doklady

Akad. Nauk USSR 78 (1951), pp.189-192. 42. B.Z. Vulikh, Some topics in the theory of partially ordered linear spaces (Russian), Izvestia

AN USSR ser. math. 17 (1953), pp.365-388. 43. B.z. Vulikh, G.Y. Lozanovskii, On representation of order continuous and regular func

tionals on partially ordered spaces (Russian), Mat. Sbomik 84 (1971), pp.331-352. 44. A.W. Wickstead, Representation and duality of mUltiplication operators on Archimedean

Riesz spaces, Compositio Math. 35 (1977), pp. 225-238. 45. K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 18 (194112), pp.

339-342. 46. A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam-New York-Oxford, 1983.


Some Aspects of the Spectral Theory of Positive Operators

XIAO-DONG ZHANG Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431, U.S.A


135

Abstract. We consider various aspects of the following problem: Let T be a positive operator on a Banach lattice such that O'(T) = {I}. Does it follow that T:::: I?

Mathematics Subject Classifications (1991): 47B65, 47AI0

Key words: positive operator, spectral theory, Banach space

Introduction

One of the basic problems in the spectral theory of bounded operators is the following: Which additional properties of a bounded operator on a Banach space would imply it to be the identity operator when its spectrum consists of the number 1 only? There are many results of this aspect. For example, if we know that the spectrum of a bounded operator T consists of the number 1 only and assume that T satisfies one of the following conditions, then T is the identity operator:

i) T is a normal operator on a Hilbert space; ii) T is an automorphism of a C* - algebra;

iii) T is an automorphism of a commutative sernisimple Banach algebra; iv) T is a double power bounded operator on Banach space, i.e., sup{IITnll:

n E Z} < 00; v) T is a lattice homomorphism on a Banach lattice.

The conclusion of i) is a well-known result. For the proofs of ii) and iii), see [1] and [11] or [12], respectively. The conclusion of iv) is a classical theorem of Gelfand's, and we refer to [7] or [13] for proofs. v) is a consequence of a theorem proved by H.H. Schaefer, M. Wolff and W. Arendt in 1978 (see [19] for the details). The original proof of this result is not elementary and uses a representation theorem for the center of a Banach lattice. C.B. Huijsmans succeeded in giving an elementary proof for this result in 1989 (see [9]). Later the author found some simple proofs for this result (see [23]) by using a wellknown theorem in the theory of entire functions of exponential type. The same

136 XIAO-DONG ZHANG

idea had been used in [7] to deduce an important property of power bounded operators on Banach spaces.

In view of v), c.B. Huijsmans and B. de Pagter proposed the following open question.

QUESTION 1 Let T be a positive operator on a Banach lattice such that iJ(T) = {I}. Is it true that T ~ I?

If this open question has a positive answer, then by applying the result to the operator T and its inverse we see that the above result v) of Schaefer, Wolff and Arendt will follow immediately. It is known that this question has a positive answer if the Banach lattice is finite dimensional. The question is still open for the general case. In this article we will consider various aspects of this open question and propose some related questions. Also we will give some important conditions under which these questions have positive answers.

1. Preliminaries

In this article we use [16] as a general reference for the theory of Banach lattices and positive operators. For the general theory of Riesz spaces we refer to [15] and [22]. We use E to denote a complex Banach lattice. Without loss of generality we may assume that E is order complete (also called Dedekind complete). L(E) denotes the Banach algebra of all bounded operators on E and Lr(E) denotes the sub algebra of all regular operators on E. It is known that Lr(E) is an order complete Banach lattice under the regular norm defined by IITllr = IIITIII if E is order complete. In this case an operator in L( E) is regular if and only if it is order bounded. Lr (E) denotes the operator norm closure of Lr (E) in L( E), and Z(E) denotes the center of E, i.e., Z(E) is the set of all operators in L(E) satisfying ITI :::; c· I for some number c ?: O. It is well-known that the center Z(E) is a closed commutative full subalgebra of L(E), algebraically and order isomorphic to the Banach algebra and Banach lattice C(X) for some compact space X. The proof for this theorem can be found in [14] and [20]. When E is order complete, Z(E) is a band of Lr(E). Denote by <P the associated band projection. Then it is easy to see that <P is contractive with respect to the regular norm. Surprisingly, it can be shown that <P is also contractive with respect to the operator norm. The following result is due to J. Voigt [21]. We will use this theorem many times later.

THEOREM 1.1 Let E be an order complete Banach lattice. Then 11<p(T)11 :S IITII for all T E Lr(E).

Since <P is a contraction with respect to the operator norm in L( E), <P can be extended to Lr(E), the operator norm closure of Lr(E) in L(E). We still use <P to denote its extension. Notice that the range of <P is Z(E).

SPECTRAL TIIEORY OF POSITIVE OPERATORS 137

2. Some Related Questions

There are some interesting questions related to Question 1 that we can ask. First we propose the following two questions.

QUESTION 2 Let T be a positive contraction on a Banach lattice such that (J(T) = {I}. Is it true that T is the identity operator?

QUESTION 3 Let T be a Markov operator on C(X), i.e., T 2: ° and Tl = 1, where X is a compact space and 1 is the constant -one function on X. Let (J(T) = {I}. Is it true that T is the identity operator?

Question 3 is due to Schaefer (see [10]). One can easily see that Question 3 is a special case of Question 2, since the condition that Tl = 1 implies that T is a contraction. Also, it is not difficult to see that if Question 1 has a positive answer, then Question 2 would have a positive answer. The proof is simple and it goes as follows. Let T 2: I and IITII :::; 1, and let A = T - I. Then A 2: ° and 0:::; nA:::; Tn for any positive integer n, so IlnAII:::; IITnll:::; 1. This implies that IIAII = 0, and so T = I.

Another interesting question that we will consider is the following:

QUESTION 4 Let T be a positive contraction such that (J(T) C {z: Izl = I} properly. Is it true that T is an isometry?

A similar question was asked in [10, p. 75] by Schaefer for positive contractions on LP-spaces, where (J(T) <: {z: Izi = I}. But in his thesis (see [24]) the author constructed an example of lattice isomorphism T on an L2-space such that:

i) T is a contraction;

ii) (J(T) = {z: Izl = I}; iii) T is not an isometry.

So the condition that (J(T) c {z: Izi = I} in Question 4 is critical. Question 4 seems more general than Question 2, but we will show in next Section that they are actually equivalent.

For the sake of completeness, we now give the counterexample mentioned in the above paragraph.

EXAMPLE 2.1 (Counterexample) Choose a continuous function g: [0, 1] --+ [0,1] as follows:

{ 1 if ° :::; x :::; 1/5 or 4/5 :::; x :::; 1,

g(x) = 1/2 if 2/5 :::; x :::; 3/5, linear otherwise.

138 XIAO-DONG ZHANG

Define a function h: R ---+ [0, 1] by

{I if -00 < x :::; 0,

( ) _ 9 ( x ) if 0 :::; x :S 1, h x - g(x _ k2 ) if k 2 :s x :s k2 + 1 (k = 1,2, ... ),

1 otherwise.

We now define T: Lp(R) ---+ Lp(R), where 1 :::; p :::; 00, by

T f(x) = h(x)f(x + 1) for all x E R.

We prove that 1) T is a contraction but not an isometry; 2) T is a lattice isomorphism such that aCT) ~ {z: I zl = I}.

From the way we define h, 1) and the first part of 2) are obvious. To prove that the spectrum of T is contained in the unit circle we need to prove that r(T-l) :::; 1. First observe that

T-n f (x) = h -1 (x - 1) h -1 (X - 2) ... h -1 (X - n) f (X - n)

So liT-nil:::; sUPxER Ih-1(x)h- 1(x + 1)··· h-1(x + n - 1)1. For any x E R, the points x, x + 1,···, x + n - 1 are contained in the interval [x, x + n - 1] of length n - 1. In [x, x + n - 1] there are at most [.;n] + 2 subintervals of the form [i, i + 1] (i is a nonnegative integer) on which the function h is not constant. Therefore, liT-nil :S 2Ifol+2. This implies that r(T-l) :S 1. We have finished the proof.

3. Main Results

Following [24], we say that a bounded linear invertible operator T on a Banach space satisfies the condition (c) if the number ° belongs to the unbounded connected component of peT). It has been proved in [24] that positive operators satisfying the condition (c) have some special properties. One of these properties is the following theorem. We will need this result in the sequel, and refer to [24] for a proof.

THEOREM 3.1 Let T be an invertible positive operator on a Banach lattice E. Assume that T satisfies the condition (c). Then there exists a positive number a and a positive integer k such that Tk 2: a . I.

Using this Theorem we can give a simple proof for the main result in [19]. For details, we refer to [24]. Also, by use of this theorem we can show that Question 2 and Question 4 are equivalent.

PROPOSITION 3.2 Question 2 is equivalent to Question 4.

SPECTRAL THEORY OF POSITIVE OPERATORS l39

Proof Let T be a positive contraction such that its spectrum is properly contained in the unit circle, then it is easy to see that T satisfies the condition (c), and so Tk ~ a· [ for some positive integer k and some positive number a by Theorem 3.1. Now let Tk = a· [ + B, where B ~ 0 and reB) = 1 - a. By the spectral mapping theorem and by the fact that the spectral radius of a positive operator is an element of its spectrum, we have

{I} ~ a-(Tk) ~ a + {z: Izl :S reB)},

and so a-(Tk) = {I}. If Question 2 has a positive answer, then Tk = J, and from which we conclude that T- 1 = T k- 1 is also a contraction. So T is an isometry. Conversely, if Question 4 has a positive answer, then Question 2 also has a positive answer, since it is only a special case of Question 4. Q.E.D.

Next we show that if the negative powers of T do not grow too fast, then the answer to Question 1 is positive. The following theorem and its proof are taken from [24]. We note that 0 :S <p(S) :S reS) . [ for any positive operator S on E. This follows from the fact that Z (E) == C (X) for some compact space X.

THEOREM 3.3 Let T be a positive operator on an arbitrary Banach lattice E such that a-(T) = {I}. If there exist 0 < a < 1/2 and a constant c ~ 0 such that

as n ~ +00,

then T ~ [. Proof We may assume that E is order complete. By assumption there exists

a bounded linear operator A E £T(E) such that Tn = exp(nA) for any integer n. In fact,

Now consider the operator valued function fez) which is given by

fez) = (exp(zA)) for all Z E C.

By Theorem 1.1 and the Remark following it, we can conclude that fez) is an entire function. Since A is a quasi-nilpotent operator, it is easy to see that fez) is an entire function of zero exponential type (see [6] for definitions).

If n is any positive integer, then Ilf(n)11 = 1I(Tn)11 :S 1 by the remark before the Theorem. So by Cartwright's Theorem (see [6, p. 180]) there exists a constant Ml such that Ilf(x)11 :S Ml for any real numbers x ~ O. If x < 0 and if x = -n + t, where n is a positive integer and 0 :S t :S 1, then it follows from Theorem 1.1 that

Ilf(x)11 = I I (T- n exp(tA))1 I :S liT-nil· II exp(tA) I I ::; M2M3 exp( ena ) :S M2M3 exp(c) exp(clxl a ),

140 XIAO-DONG ZHANG

where Mz = sUPO<t<l II exp(tA) I! and M3 is given by the condition in the Theorem. Therefore~ there exists a constant M such that

IIJ(x)11 ~ M exp(clxla ) for all real numbers x.

Now by [6, Thm. 6.6.9, p. 97] there are constants K and b such that

IIJ(z)II ~ K exp(blzl a ) for all complex numbers z.

Next we consider the entire function g(z) = J(zZ). It is easy to see that for any E > 0 there exists a constant L depending on E such that

Ilg(z)II ~ Kexp(blzl za ) ~ Lexp(Clzl)

since 2a < 1 by assumption. Moreover Ilg(±n)11 = IIJ(nZ)II = II<1>(Tn2)II ~ 1. So it follows from a well-known theorem in the theory of entire functions (see [6, p. 183]) that g(z) is a constant function, and so J(z) is a constant function. In particular, we have J(O) = J(l), i.e., (T) = I. Therefore, T 2: I. Q.E.D.

COROLLARY 3.4 Let T be a positive operator on a Banach lattice such that a(T) = {1}. If there exists a positive number k such that

as n ---t +00,

then T 2: I. Conversely, if T 2: I, then (Tn) = I for every integer n. Proof The proof is similar as above. Using the same notations, we consider

the entire function J(z) given above. Then by assumption we have

IIJ(±n)I! = II (T±n) II = O(nk).

Now it follows from [6, Thm. 10.2.11, p. 183] that J(z) is a polynomial of degree not exceeding [k] + 1. Observe that IIJ(n)II = II<1>(Tro)II ~ 1 for every positive integer n. So J(z) must be a constant function. In particular we obtain (T) = I. So T 2: I.

Conversely, if T ;::: I, then T = I + A for some positive quasi-nilpotent operator A. Since any positve quasi-nilpotent operator belongs to Z(E)d, by the Remark before Theorem 3.3 we see that (Tn) = I for any positive integer n. Now T- 1 = I + I:~l(-I)kAk and I:k=l Ak is a positive quasi-nilpotent operator, so (T-n) = I for any positive integer n. Q.E.D.

REMARK 3.5 It follows from Theorem 1.1 that the condition in Corollary 3.4 will be satisfied if liT-nil = O(nk). This is the case when T- 1 is a power bounded operator. Moreover, if 1 is a pole of the resolvent of T, then it is well known that IIT±nl! = O(nk) for some positive integer k. So we obtain the following assertion.

COROLLARY 3.6 Let T be a positive operator such that a(T) = {I}. If 1 is a pole oj the resolvent oj T, then T 2: I.

SPECTRAL TIIEORY OF POSITIVE OPERATORS 141

REMARK 3.7 The invariant subspace problem for those operators on Banach spaces that satisfy the growth condition in Theorem 3.3 has been studied in [4]. Some of the ideas used in the proofs of the above results are borrowed from this paper.

THEOREM 3.8 Let T be a positive contraction on a Banach lattice E such that a(T) c {z: I z I = 1} properly. If T satisfies one of the following conditions, then T is an isometry.

i) The condition in Theorem 3.3 or in Corollary 3.4; ii) T is a lattice homomorphism.

In particular, if a-(T) = {I}, then T is the identity operator on E. Proof It follows from Theorem 3.1 that a-(TN) = {I} for a positive integer

N. Now notice that TN also satisfies the growth condition in Theorem 3.3 or in Corollary 3.4. So, by Theorem 3.3 or Corollary 3.4, we have TN 2: I. Now let TN = I + A. Then A is a positive operator on E. Since T is a contraction, we have IlnA11 ::; II (I + A)n II ::; 1 for any positive integer n. So A = 0 and TN = I. Thus T- 1 = TN -1 , from which we see that II T- 111 ::; 1. So T is an isometry. If T is a lattice homomorphism, then (J(TN) = {I} implies that TN = I. Q.E.D.

REMARK 3.9 Theorem 3.8 was obtained in [18] by Schaefer for finite dimensional V -spaces.

Finally, we mention an interesting consequence of Corollary 3.6. It follows immediately from Corollary 3.6. The following elegant proof is due to F. Beukers. The author would like to take this moment to thank C.B. Huijsmans for informing him of this proof.

THEOREM 3.10 If T is a positive operator on a finite dimensional Banach lattice such that a-(T) = {I}, then T 2: I.

Proof Let dim(E) = n. Then T can be represented as an n x n matrix with nonnegative entries. Let A = T - I. Then A 2: -I. Now let A be represented as an n x n matrix (aij )nx n. Then aij 2: 0 for i # j and aii 2: -1.

Consider A2. Since (J(A2) = {O}, the trace of A2 is zero. But the trace of A2 is the sum of its diagonal elements, all of which are nonnegative. In fact, the i-th diagonal element of A2 is Lk=1 aikaki and each aikaki is nonnegative. Therefore, all the diagonal elements of A2 are zero. In particular, we obtain aii = O. Hence all the entries of A are nonnegative and so T - I 2: O. Q.E.D.

References

1. C.A. Akemann, P.A. Ostrand, The spectrum of a derivation of a C* -algebra, 1. London Math. Soc. (2) 13 (1976), pp. 525-530.

2. W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. 1. 32 (1983), pp. 199-215.

142 XIAO-DONG ZHANG

3. W. Arendt, D.R. Hart, The spectrum of quasi-invertible disjointness preserving operators, J. Funct. 68 (1986), pp. 149-167.

4. A. Atzmon, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144 (1980), pp. 27-63.

5. B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland, Amsterdam, 1988.

6. R.P. Boas, Jr., Entire Functions, Academic Press, 1954. 7. J. Esterle, Quasimultipliers, representations on H oo , and the closed ideal problem for

commutative Banach algebras, in: Radical Banach Algebras and Automatic Continuity, Lecture Notes in Math. 975 pp. 66-162.

8. C.B. Huijsmans, Elements with unit spectrum in Banach lattice algebras, Indag. Math. 50 (1988), pp. 43-45.

9. C.B. Huijsmans, An elementary proof of a theorem of Schaefer, Wolff and Arendt, Proc. Amer. Math. Soc. 105 (1989), pp. 632-635.

10. From A to Z, Proceeding of a Symposium in Honour of A.C. Zaanen: Edited by c.B. Huijsmans, M.A. Kaashoek, W AJ. Luxemburg and W.K. Vietsch: Mathematical Center Tracts 149 , Amsterdam, 1982.

11. B. Johnson, Automorphisms of commutative Banach algebras, Proc. Amer. Math. Soc. 40 (1973), pp. 497-499.

12. H. Kamowitz, S. Scheinberg, The spectrum of automorphisms of Banach algebras, J. Funct. Analysis 4 (1969), pp. 268-276.

13. Y. Katznelson, L. Tzafriri, On power-bounded operators, J. Funct. Analysis 68 (1986), pp. 313-328.

14. W.A.J. Luxemburg, Some aspects of the theory of Riesz spaces, University of Arkansas, Lecture Notes in Mathematics, Fayetteville 4 (1979).

15. W.A.J. Luxemburg, A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam, 1971. 16. H.H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, 1974. 17. H.H. Schaefer, On the O-spectrum of order bounded operators, Math. Z. 154 (1977), pp.

78-84. 18. H.H. Schaefer, On positive contractions in LP -spaces, Trans. Amer. Math. Soc. 257

(1980), pp. 261-268. 19. H.H. Schaefer, M. Wolff, W. Arendt, On lattice isomorphisms and the groups of positive

operators, Math. Z. 164 (1978), pp. 115-123. 20. A.R. Schep, Positive diagonal and triangular operators, J. Operator Theory 3 (1980),

pp.165-178. 21. J. Voigt, The Projection onto the center of operators in a Banach lattice, Math. Z. 199

(1988), pp. 115-117. 22. A.C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983. 23. X.D. Zhang, Two simple proofs for a theorem of Schaefer, Wolff and Arendt, Caltech,

Pasadena, California, August, 1990. Submitted. 24. X.D. Zhang, On Spectral properties of positive operators, Ph. D. Thesis, California Institute

of Technology, Pasadena, California, USA (April, 1991).


Problem Section

PROBLEM 1 (Y.A. Abramovich)

143

It is well known [1] (see also [5] for a rather simple proof) that if T is a band preserving operator on a Banach lattice X, then T is automatically continuous. However, if T is only a disjointness preserving operator from a Banach lattice X into a Banach lattice Y, then T may be discontinuous [1], [2] even if X and Y are Dedekind complete.

QUESTION 1 Assume that T: X --+ Y is additionally invertible. Does this imply that T is automatically continuous?

In the special case when X = C(K1) and Y = C(K2), where K 1, K2 are compact Hausdorff spaces, the answer to the previous question is 'YES' and it may be found in [4].

Now let T be a one-to-one disjointness preserving operator from a Banach lattice X onto a Banach lattice Y.

QUESTION 2 Does T- 1 preserve disjointness?

This question is partly motivated by the main result in [3], asserting that for each positive isometric operator T from a Banach lattice X onto a Banach lattice Y its inverse T-I is necessarily positive.

Of course, if the answer to Question 1 is yes, then it would imply a positive answer to Question 2. If the answer to Question 2 is positive, then it is interesting to answer the same question for arbitrary vector lattices.

I. Y.A Abramovich, A.I. Veksler, A. V. Koldunov, Operators preserving disjointness, Dokl. Akad. Nauk USSR 248 (1979), 1033-1036.

2. Y.A Abramovich, Multiplicative representation of operators preserving disjointness, Nederl. Akad. Wetensch. Proc. Ser. A 86 (1983), 265-279.

3. Y.A. Abramovich, Some results on the isometries in normed lattices, Optimization, Novosibirsk 43 (1988), 74-80.

4. K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), 139-144.

5. AW. Wickstead, Representations of Archimedean Riesz spaces by continuous functions, this volume.

Editorial note. In the meantime the following result has been proved by C.B. Huijsmans and B. de Pagter: an invertible disjointness preserving operator from a uniformly complete vector lattice onto a normed vector lattice is necessarily order bounded and has therefore a disjointness preserving inverse. Particularly, any bijective disjointness preserving operator between two Banach lattices is continuous (norm bounded).

144 PROBLEM SECTION

PROBLEM 2 (Y.A. Abramovich, A. W. Wickstead) It is proved in [1] that any order bounded operator T from a separable Banach

lattice X = C(K)-space into a Cantor Banach lattice Y is regular.

QUESTION 1 Does this result remain true for an arbitrary separable Banach lattice X?

Ifthe answer to this Question is affirmative, then it would be interesting to find out whether we can replace order bounded operators by continuous operators? Both of these versions are motivated also by investigations in [2], where many other examples of concrete spaces as presented for which any order bounded or/and continuous operator is regular.

1. y.A. Abramovich and A.W. Wickstead, The regularity of order bounded operators into C(K), II. Submitted.

2. Y.A. Abramovich and A.W. Wickstead, Regular operators from and into a small Riesz space, Indag. Math. New Series 2 (1991), 257-274.

PROBLEM 3 (Y.A. Abramovich, A. W. Wickstead) It has been known since [2] that a compact linear operator from one Banach

lattice into another need not have a modulus. The example given there is not even regular, but in [1] we gave an example of a regular compact operator which fails to have a modulus.

QUESTION 1 If a compact operator is dominated by a positive compact operator, must it have a modulus? If so, must that modulus be itself compact?

1. Y.A. Abramovich, A.W. Wickstead, A compact operator without a modulus, Proc. Amer. Math. Soc. (to appear).

2. U. Krengel, Remark on the modulus of compact operators, Bull. Amer. Math. Soc. 72 (1966), 132-133.

PROBLEM 4 (Y.A. Abramovich, A. W. Wickstead) Let X, Y be two Archimedean Riesz spaces and let U(X, Y) (Lb(X, Y))

denote the space of all regular (order bounded) operators between X and Y. Assume that Lr (X, Y) is a Riesz space.

QUESTION 1 Is it true that U(X, Y) = Lb(X, Y)?

This question was also asked in [1].

1. Y.A. Abramovich, A.W. Wickstead, Regular operators from and into a small Riesz space, Indag. Math. New Series 2 (1991), 257-274.

PROBLEM 5 (Y.A. Abramovich, C.D. Aliprantis, O. Burkinshaw)

PROBLEM SECTION 145

Recall that a Krein space is a Banach space partially ordered by a closed cone having a strong unit. It is shown in [1, Example 4.6] that the second dual of a Krein space E need not be a Krein space.

Find necessary and sufficient conditions on a Krein space E to guarantee that E" is likewise a Krein space.

1. Y.A Abramovich, C.D. Aliprantis, O. Burkinshaw, Positive operators in Krein spaces, this volume.

PROBLEM 6 (V.A. Abramovich, C.D. Aliprantis, O. Burkinshaw) Let Kl, K2: E ~ E be two positive compact operators on a Dedekind com

plete Banach lattice.

QUESTION 1 Is the supremum operator Kl V K2 (or, equivalently, the infimum operator Kl 1\ K 2) compact?

HISTORY AND COMMENTS

We observe that 0 ::; Kl V K2 ::; Kl + K 2•

a) There is a compact regular operator on £2 whose modulus is not compact; U. Krengel [7].

b) If Kl and K2 are finite rank operators, then Kl V K2 is necessarily compact [4, p. 272], but in general, Kl V K2 is not a finite rank operator [6], [8].

c) It is conjected in [1] that Kl V K2 should exist even if the Banach lattice E is not Dedekind complete. On the other hand, an example is presented in [1] of a compact regular operator T on a Banach lattice E such that T does not have a modulus (and hence, necessarily, E is Dedekind incomplete).

d) If E' and E have order continuous norms, then Kl V K2 is compact; P.G. Dodds and D.H. Fremlin [5].

e) (Kl V K 2)3 is always a compact operator; C.D. Aliprantis and O. Burkinshaw [2].

f) If E' or E has order continuous norm, then (Kl V K2f is a compact operator; C.D. Aliprantis and O. Burkinshaw [2].

g) (Kl V K2)2 is always a weakly compact operator; C.D. Aliprantis and O. Burkinshaw [3].

h) If either E' or E has order continuous norm, then Kl V K2 is weakly compact; A.W. Wickstead [9].

i) It may be easier to construct first two positive compact operators Kl, K2 from a Banach lattice E into a Dedekind complete Banach lattice F such that Kl V K2 is not compact.

1. Y.A Abramovich, AW. Wickstead, A compact regular operator without a modulus, Proc. Amer. Math. Soc., forthcoming.

2. C.D. Aliprantis, O. Burkinshaw, Positive compact operators on Banach lattices, Math. Z. 174 (1980), 289-298.

146 PROBLEM SECTION

3. C.D. Aliprantis, O. Burkinshaw, On weakly compact operators on Banach lattices, Proc. Amer. Math. Soc. 83 (1981), 573-578.

4. C.D. Aliprantis, O. B urkin shaw, Positive Operators, Academic Press, New York, 1985.

5. P.G. Dodds, D.H. Fremlin, Compact operators in Banach lattices, Israel 1. Math. 34 (1979), 287-320.

6. G. Godefroy, Sur les operateurs compact reguliers, Bull. Soc. Belgium 30 (1978), 35-37.

7. U. Krengel, Remark on the modulus of compact operators, Bull. Amer. Math. Soc. 72 (1966), l32-133.

8. H.U. Schwarz, Banach Lattices and Operators, Teubner Texte 71, Teubner, Leipzig, 1984.

9. A.W. Wickstead, Extremal structure of cones of operators, Quart. 1. Math. 04'ord 32 (1981), 239-253.

PROBLEM 7 (Y.A. Abramovich, C.D. Aliprantis, W.AJ. Luxemburg) If E and F are arbitrary Banach lattices, then we denote by L+(E, F) the

cone of all positive continuous operators from E to F. The subcone of all positive compact operators is denoted by K+(E, F). Both L+(E, F) and K+(E, F) are equipped with the order inherited from the space U(E, F) of all regular operators. Note that U(E, F) is a Riesz space if L+(E, F) is a lattice if L+(E, F) is an upper (or lower) lattice.

QUESTION 1 Does there exist a pair of Banach lattices E and F such that 1) L+(E, F) and K+(E, F) are both lattices, but K+(E, F) is not a sublattice

of L+(E, F); or 2) L+(E, F) is not a lattice, but K+(E, F) is?

PROBLEM 8 (Y.A. Abramovich, CoD. Aliprantis, W.AJ. Luxemburg) Let us call two operators T 1, T2 : E -t F strongly disjoint if their ranges are

disjoint, i.e., if 1 T1xl 1 /\ 1 T2X21= 0 for all Xl,X2 E E. We call S E U (E, F) an elementary operator if S = A - B for strictly

disjoint positive operators A and B. Clearly, any elementary operator S has a modulus 1 S 1 and 1 S 1= A + B.

QUESTION 1 Does there exist a pair of Banach lattices E and F such that the only regular operators T: E -t F with a modulus are those of the form T = Tl + T2, where Tl is a finite rank operator, T2 is an elementary operator and Tl and T2 are strongly disjoint?

PROBLEM 9 (C.D. Aliprantis, O. Burkinshaw) We repeat here an old problem proposed a few years ago in [1].

PROBLEM SECTION 147

QUESTION 1 Does every positive compact operator between Banach lattices factor (with positive compact factors if possible) through a reflexive Banach lattice?

HISTORY AND COMMENTS

a) It is known that every compact operator between Banach spaces factors with compact factors through a separable reflexive Banach space; T. Figiel [4], W.B. Johnson [5].

b) Every weakly compact operator between Banach spaces factors through a reflexive Banach lattice; W.J. Davis, T. Figiel, W.B. Johnson and A. Pelczynski [3].

c) A positive weakly compact operator between Banach lattices need not factor through a reflexive Banach lattice; M. Talagrand [7].

d) The square of a positive compact (resp. weakly compact) operator on a Banach lattice factors through a reflexive Banach lattice with positive compact (resp. weakly compact) factors; C.D. Aliprantis and O. Burkinshaw [1].

e) the problem has a positive answer if we assume that the spaces satisfy the approximation property; Z.L. Chen [2].

f) Some additional results about possible positive answers can be found in the paper by I. M. Popovici and D.T. Vuza [6].

1. C.D. Aliprantis, O. Burkinshaw, Factoring compact and weakly compact operators through reflexive Banach lattices, Trans. Amer. Math. Soc. 183 (1984), 369-381.

2. Z.L. Chen, The factorization of a class of compact operators, Dongbei Shuxue 6 (1990), 303-309.

3. W.J. Davis, T. Figiel, W.B. Johnson, A. Pe1czynski, Factoring weakly compact operators, 1. Funct. Anal. 17 (1974), 311-327.

4. T. Figiel, Factorization of compact operators and application to the approximation property, Studia Math. 45 (1973), 191-210.

5. W.B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345.

6. I.M. Popovici, D.T. Vuza, Factoring compact operators and approximable operators, Z. Anal. Anwendungen 9 (1990), 221-233.

7. M. Tal agrand , Some weakly compact operators between Banach lattices do not factor through reflexive Banach lattices, Proc. Amer. Math. Soc. 96 (1986). 95-102.

PROBLEM 10 (W. Arendt, G. Greiner) The A = 0 problem for positive Co-groups.

QUESTION 1 Let A be the generator of a positive Co-group on a Banach lattice E. If (T(A) = {O}, does it follow that A = O?

148 PROBLEM SECTION

By the results of [3] the answer is positive if E is an AM-or an AL-space, but the proof is based on a representation theorem valid on E = C (K), which shows that (after a similarity transformation) A is the additive perturbation of the generator of an automorphism group by a multiplication operator. So the proof of the special cases does not give an indication for the general case.

On the other hand, the analogous discrete version is the so-called T = I problem, which has a positive answer.

THEOREM lfT is a lattice isomorphism and <J(T) = {I}, then T = I.

This had first been proved in [2]. Simpler proofs are given in [1] and by Huijsmans and De Pagter in the present volume.

The problem raised here is of interest in the context of the (weak) spectral mapping theorem for Co-semigroups. By this one understands the formula

<J(T(t)) = expt<J(A) (t > 0) (1)

for the generator A of a Co-semigroup (T( t) )t;:::o. The spectral mapping theorem does not hold, in general (we refer to [5] for a detailed discussion). However, in [3] it was shown to hold for generators of positive groups if E is an AM-or an AL-space. Thus, in this case, the positive solution of the Problem follows from its discrete version, the above theorem.

Without positivity assumptions the situation is the following. There exists a Co-group (U(t))tER on a Banach space G with generator B such that dB) = 0 (see [4, Sec. 23.16, p. 665]). This shows that (1) is violated. Taking the direct sum A = B EB 0 on G EB C one obtains the generator A of a group such that <J(A) = {O} but A i= O.

REMARK The first phenomenon cannot happen for positive groups. Their generators always have nonempty spectrum (see [5, C-III, Cor. 1.6]).

For polynomially bounded Co-groups the spectral mapping theorem holds (see [5, A-III, Thm. 7.4]). Thus, again in view of the Theorem, the problem has a positive answer if in addition one assumes that the group is polynomially bounded.

1. W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. 1. 32 (1983), 199-215.

2. H.H. Schaefer, M. Wolff, W. Arendt, On lattice isomorphisms and groups of positive operators, Math. Z. 164 (1978), 115-123.

3. W. Arendt, G. Greiner, The spectral mapping theorem for one-parameter groups of positive operators on Co(X), Semigroup Forum 30 (1984), 297-330.

4. E. Hille, R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. ColI. Publ. 31, Amer. Math. Soc., Providence (R.I.), 1957.

5. R. Nagel (ed.), One Parameter Semi-Groups of Positive Operators, Lecture Notes in math. 1184, Springer, Berlin, 1986.

PROBLEM SECTION 149

PROBLEM 11 (C.B. Huijsmans, B. de Pagter)

QUESTION 1 Let E be a Banach lattice, T: E --t E a positive linear mapping such that (J( T) = {1}. Does this imply T 2:: I?

The answer is yes in the following cases: a) dimE < 00; b) T m E Z(E), the centre of E, for some mEN; c) T is a lattice homomorphism; d) 1 is a pole of the resolvent.

It is even unknown whether T 2:: 0, (Jo(T) = {1} «Jo(T) is the order spectrum) implies T 2:: I.

QUESTION 2 Let E be a Banach lattice, ° ::; T: E --t E such that T 1\ I = 0. Does this imply r(I - T) 2: I?

(Same question if E is Dedekind complete and T E Lb(E) satisfies T ..lI). The answer is affirmative in the following cases:

a) dimE < 00; b) T m E Z(E) for some mEN; c) Tn 11 for all n EN; d) T is a lattice homomorphism.

PROBLEM 12 (W.AJ. Luxemburg) According to the celebrated theorem of R.C. James [1], a Banach space E is

reflexive if and only if every continuous linear functional attains its maximum on the unit ball of E.

Although the proof of this theorem has been considerably simplified over the years (see [2]) it still remains a difficult theorem to prove, reflecting the depth of the theorem. For this reason one may ask whether the corresponding theorem for Banach lattices or (saturated) Banach function spaces may be more susceptibel of an easier approach? That this is perhaps the case is suggested by the following simple observations.

If E is a Banach lattice and if ¢ is a continuous linear functional on E (¢ E E'), where E' is the dual space of E, then ¢ admits a Hahn decomposition, i.e., there exist elements ° ::; XI,X2 E E satisfying inf(xI,x2) = 0, such that ¢(XI - X2) = II¢II and ¢ = ¢+ on the ideal generated by Xl and ¢ = ¢- on the ideal generated by X2.

Indeed, if X E E satisfies Ilxll = 1 and ¢(x) = Ilxll, then from II¢II = II 1</>111 it follows that (¢+ - ¢-)(x+ - x-) 2: (¢+ + ¢-)(x+ + x-), and so ¢+(x-) + ¢-(x+) ::; 0. Hence, ¢+(x-) = ° and ¢-(x+) = 0, and the result follows. From this simple fact we may conclude immediately that if the band E~ of the order continuous linear functionals on E is total, i.e., 0 (E~) = {x E E: 1¢1(lxi) = 0, ¢' E E~} = {O}, then the norm of E is order continuous whenever

150 PROBLEM SECTION

every functional ¢ E E' obtains its maximum on the unit ball of E. To see this one only needs to take ¢ to be the difference of a positive order continuous linear functional and a positive singular linear functional. In particular, it follows in an elementary way that if E is a (saturated) Banach function space with the James property, then the function norm is order continuous. To use this kind of argument to prove reflexivity runs into difficulties.

Finally, we note that in James' theorem for Banach lattices we cannot restrict the condition to hold for positive linear functionals only. The space L1 [0, 1]) of the Lebesgue-integrable functions is a counterexample.

1. R.C. James, Reflexivity and the supremum of linear functionals, Ann. of Math. 66 (1957), 159-169.

2. K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer, Berlin, 1980.

PROBLEM 13 (L. Maligranda) Let X, Y be Banach lattices and T E LeX, Y). There exists a constant G E

[1,2] such that

11((Tx)2 + (Ty)2) ~ Ily ::; GIITII . II(x2 + y2) ~ Ilx for all x, y E X. It was shown by J.L. Krivine that G ::; -12. Moreover, if T 2:: 0, then G = 1.

It is known that G = 1 as well in the case X = Y = Lp.

QUESTION 1 Is it always true that G = 1 in the case X = Y?

PROBLEM 14 (J. Martinez) Suppose that G is a lattice-ordered group and H is an i-subgroup. We say that

G is a strongly rigid extension of H (or that H is strongly rigid in G) if for each 0::; g E G there is h 2:: g, h E H so that h·LL = gJ..J... The following paraphrases [1, Prop. 1]: If H is strongly rigid in G, then the map P I-t PnH is a surjection from Spec(G), the set of prime convex i-subgroups of G, onto spec(H), such that if P and Q are incomparable prime i-ideals of G, then their contractions are also incomparable. In particular, this same contraction is a homeomorphism between the minimal primes.

We pose the following question, in the context of commutative f-algebras with 1, which satisfy the bounded inversion condition, that is, a > 1 implies that a is invertible.

QUESTION 1 If A is an f-algebra, when does it have a minimal strongly rigid f-subalgebra (which also satisfies the bounded inversion)?

For example, it is easy to show that any strongly rigid f-subalgebra of A must contain all the idempotents of A. If A = G(X), and X is a compact Hausdorff

PROBLEM SECTION 151

space which is basically disconnected, then S(X), the subalgebra generated by the idempotents of A, is the unique minimal strongly rigid J-subalgebra of A.

Two intriguing special cases of the above question remain unanswered:

QUESTION 2 a) IfC(X) (X compact, Hausdorff) has a unique minimal strongly rigid J-subalgebra, then is X basically disconnected?

b) Which J-algebras have no proper strongly rigid J-subalgebras?

For example, if aN denotes the one-point compactification of N, then does C(aN) have a proper strongly rigid J-subalgebra?

The proposition quoted from [1] brings some evidence to the conjecture that the answer to the second part of b) is 'NO', making the answer to a), likewise, negative. For instance, in search for an J-subalgebra B of C(X) (again X compact and Hausdorff) which is strongly rigid, the map M ~ M n B is a homeomorphism between the maximal ideals of the two algebras. This means that B is uniformly dense in C (X).

1. P. Conrad, 1. Martinez, On adjoining units to hyperarchimedean £-groups, to appear.

PROBLEM 15 (P. Meyer-Nieberg)

QUESTION 1 Which positive operators factor approximately through an Mspace?

Suppose that E is an L-space and that T: LI (/1) ---+ E is weakly compact. It is well-known that T is representable in the sense that

TJ = J Jgd/1

where g E L oo (/1, E) satisfying get) E Thall(L1(/1)) for almost all t. Since Tball(L1(/1)), is a relatively weakly compact subset of an L-space it is approximately order bounded. Thus, for every E > 0 there is Z E E+ (depending on c) such that

Thall(L1(/1)) C [-z, z] + cban(E).

We define Tc;: L 1 (/1) ---+ E by

Td = J J(g+ 1\ Z - g- 1\ z) djJ,.

Clearly, Tc; is a bounded linear operator satisfying

152 PROBLEM SECTION

Obvious, TE has a factorization

TE : L1(JL) -+ Ez ~ U'O(v) '----+ E.

Moreover, the construction of TE yields that II T - TE II :'S E. The same construction can be done if T: L1 (v) -+ E is an L-weakly compact operator into an arbitrary Banach lattice E.

QUESTION 2 Assume that E and F are Banach lattices with order continuous norms. Give a characterization of (positive) linear operators T: E -+ F which can be approximated in the operator norm by operators TE which positively factor through an M -space.

It is clear that every operator with this property is L-weakly compact. Moreover, it is a Dunford-Pettis operator. The converse, however, need not to be true. The natural embedding V(O, 1) '----+ U(O, 1), where 00 > p > r 2: 1, is Lweakly compact but fails to be a Dunford-Pettis operator. For further information see [1].

1. P. Meyer-Nieberg, Ueber approximativ faktorisierbare Operatoren, Arch. Math. 29 (1977), 549-557.

PROBLEM 16 (A.R. Schep)

QUESTION 1 Let E be a Banach lattice with indices s(E) and a(E). Does the following hold.

1

1) P < s(E) if and only if n- p II Vk=l IXkll1 -+ ° for all norm bounded disjoint sequences {Xk}.

1

2) p > a(E) if and only if L~=l n -prllxnll < 00 for all positive summable disjoint sequences {xn }.

REMARK It was proved in [1] that the above characterizations of the indices of a Banach lattice are true if one removes the disjointness hypotheses.

1. Anton R. Schep, Krivine's theorem and the indices of a Banach lattice, these proceedings.

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Positive Operators and Semigroups on Banach Lattices: Proceedings of a Caribbean Mathematics Foundation Conference 1990