BASIC SEQUENCES AND SUBSPACES IN LORENTZ SEQUENCE …

transactions of theamerican mathematical societyVolume 263, Number 2, February 1981

BASIC SEQUENCES AND SUBSPACES IN LORENTZ SEQUENCE

SPACES WITHOUT LOCAL CONVEXITY

BY

1NICOLAE POPA

Abstract. After some preliminary results (§ 1), we give in §2 another proof of the

result of N. J. Kalton [5] concerning the unicity of the unconditional bases of lp,

0<p< 1.Using this result we prove in §3 the unicity of certain bounded symmetric block

bases of the subspaces of the Lorentz sequence spaces d(w,p), 0 > 1 there are subspaces of d(w,p), 0 <p < 1, which contain

no complemented subspaces of d(w, p) linearly homeomorphic to lp. In fact there

are spaces d(w,p), 0 <p < 1, which contain no complemented subspaces linearly

homeomorphic to lp. We conjecture that this is true for every d(w,p), 0 <p < 1.

The answer to the previous question seems to be important: for example we can

prove that a positive complemented sublattice E of d(w, p), 0 <p < I, with a

symmetric basis is linearly homeomorphic either to lp or to d(w,p); consequently, a

positive answer to this question implies that E is linearly homeomorphic to d(w, p).

In §5 we are able to characterise the sublattices of d(w,p), p — k~' (however

under a supplementary restriction concerning the sequence (wn)J°_,), which are

positive and contractive complemented, as being the order ideals of d(w, p).

Finally, in §6, we characterise the Mackey completion of d(w, p) also in the case

p = k~\ *EN.

1. Preliminary results. Let Ibea real linear space and 0 <p < 1. A function,

denoted by || ||, defined on X with the values in R+, is called ap-norm (or briefly a

norm) if the following conditions are verified.

1. ||x|| = 0 if and only if x = 0.

2. ||ax|| = \a\p||;c|| for x G X and a G R.

3. H* + y\\ < \\x\\ + \\y\\ for x,y G X.Then the subsets U„ = {x G X: \\x\\ < n~x), for n G N, constitute a fundamen-

tal system of neighbourhoods of zero for a metric linear topology of X. If X is

complete with respect to this topology we say that X is ap-Banach space.

A sequence (x„)^°_i in X is called a basis if for every x G X there is a unique

sequence of scalars (a„)^x such that x = 2°1, a¡x¡.

The following lemma is essentially known (see Theorem III.6.1, Theorem 6.5 of

[10]):

Lemma 1.1. Let (xi)f=x be a sequence in X. The following assertions are equivalent:

1. 77ie series S".| •*„(„) converges for every permutation v of the integers.

Received by the editors September 13, 1979 and, in revised form, December 4, 1979.

AMS (MOS) subject classifications (1970). Primary 46A45, 46A10; Secondary 46A35.Key words and phrases. />-Banach spaces, symmetric bases, complemented subspaces.

'Supported by Alexander von Humboldt Foundation.

© 1981 American Mathematical Society

0002-9947/81/0000-O058/SO7.50

431

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

432 NICOLAE POPA

2. The series 2,e/) x¡ converges for every subset A c N.

3. For every e > 0 there exists an integer n such that ||2ie//Jc,|| < efor every finite

set of integers 77 which satisfies min{< G 77} > n.

4. For any bounded sequence of real numbers (aB)"_1; the series SjLj aixi con-

verges, when 2°!, xi converges.

The proof will be omitted.

A basis (xn)^=x in X is said to be unconditional if for every X ^ x = ~2*LX a¡x,,

the sequence (anxn)™=, verifies one of the equivalent assertions of Lemma 1.1.

If 0 < inf„ ||x„|| < supj|jcj| < + oo, we say that the basis (xn)^_x is bounded.

The following corollary is also known.

Corollary 1.2. Let (x¡)^Lx be an unconditional bounded basis of X. Then

x\\\ = supIM 0. Then there

exists the sequence of integers (nk)kx>=x so that |||jcn — xn'||| < e for n > nx and

Ujjfifc _ x"*+i||| < e/2* for every k G N. Since (X, || ||) is a /7-Banach space and

||jc|| < 111*111, there exists x = Sn°_,(x','+' - x"") + x"' G X and |||jc

for n > nx. □

Corollary 1.4. There exists a constant 0 < M < +oo such that

2 a>bixii=i

< M 2 «,•*,■í=isup|è,|'

< 2í

(1.2)

where (x,-)," , is an unconditional bounded basis of X and x = 2," x a¡x¡ G X.

Proof. It follows by Lemma 1.3 and by the open mapping theorem. □

Two bases (xn)™=, and (y„)^ x of X are equivalent and we write (x„) — (yn), if

for every sequence of scalars (an)™=x, S^L, anxn converges if and only if S^ anyn

converges. A basis (xn)™= x of X is called symmetric if every permutation (x^n))^_,

of (xn)™=, is a basis of X equivalent to (xn)™_,.

Let w = (w,-)," , G c0\ /,, where 1 = w, > w2 > • • • > w„ > • • • > 0. For a

fixed/7 with 0 ) = \a = (a,.)", G c0: \\a\\p,w = sup 2 |äWof ' w¡ < +0°I *■ i = i

where 77 is an arbitrary permutation of the integers.

Then X = (d(w,p), || || ) is a/7-Banach space and the canonical basis (xn)"_,, is

a symmetric basis of X (see [4]). If p > 1 we can define analogously d(w,p), which

is a Banach space under the norm

(00 \ i//>

2 I «„(,/ ■ w, , a - (a,)"., e </(w, />).1 = 1 /


LORENTZ SEQUENCE SPACES 433

A sequence (y„)^ x in a /7-Banach space with a basis (x„)"_, is called a block

basic sequence of (x„)^°_,, if there is an increasing sequence of integers (p„)^x such

that.y„ = 2f_+;„ + i <*iX, with (aX-i scalars.

It is known that (>»„)"_, is a basis of Sp{.y„: n G N} (see II.5.6 of [10]).

If E and F are two Banach spaces and 0 < r < + oo, we say that the linear

bounded operator T: E —» F is r-absolutely summing if, for any finite set of

elements (jc,)"_ , of 7s, there is c > 0 such that

(¿ll/Xlf) <c-supH 2k(*,)f) : *'G £', ||*'|| < lj. (1.3)

Denote the set of all r-absolutely summing operators from TJ to F by Pr(E, F).

If T G Pr(E, F) and r > 1 then wr(T) = inf{c > 0: c verifies (1.3)} is a norm on

Pr(E, F), which is a Banach space with respect to this norm.

We recall here two more theorems which we need:

If A is an infinite matrix of real numbers (ay-)"_, and 0 < p < 1, then we denote

by

IM || oo j, = SUp'(2<1\ 7

AÍJ ajkXk

i/P

where \\x\\x = supteN|^|.

Theorem 1.5 (see Theorem 2 of [2]). Let 0 <p < 1. For every matrix A such

that ||j4||w. < oo, we have the inequality

mf{\\d\\p/{X.p)-\\B\\xy.A = (diagrf) ° B) < K\\A\\X,P (1.4)

where

2KAI"'})

diag í/ ù //ie matrix which has on the diagonal the numbers dn, and K is a positive

constant depending only on p.

Theorem 1.6 (see Theorem 94 of [9]). Let (X, p) be a measure space and H a

Hilbert space. Then any linear bounded operator from LX(X, p) on H is p-absolutely

summing for all 0 < p < oo.

(In fact Maurey proved a stronger version of Theorem 1.6.)

2. The unicity of the unconditional bases of lp, 0 <p < 1. In this section we give a

new proof of Kalton's result [5] concerning the unicity of unconditional bases of lp,

0 < p < 1. The proof follows the idea of Lindenstrauss and Pelczyñski's proof

concerning the unicity of unconditional bases of /, [7].

Theorem 2.1. Any two unconditional bounded bases of lp, 0 <p < 1, are equiva-

lent.

Proof. If (e„)"_, is the canonical basis of lp, 0 <p < 1, let (x„)"_„ where

x, = 2Jli b„e. such that 2JLi|6yK = 1 for every /' G N, another unconditional


434 NICOLAE POPA

(assumed normalized) basis and let^ = 2°1, aixi an element of lp, 0 lp be the operator defined by the infinite matrix A = (a¡by)°°_x. By

(1.2) it follows that A is a continuous linear operator, and moreover

ooj, = sup

W < i2 \atxt

!//>A/ 2 a,*,

!//>(2.1)

Theorem 1.5 shows us that for every e > 0 there are K = K(p) > 0, the diagonal

matrix D = (didy)?J=x, where d = (4)», G lp/{1_p) and the matrix C = (cy)°°_,

such that

4 - D ■ C,

||C||oo,i - sup

I\I<1

CO / CO \

2 2 V/, k7=1Vi=l /

< +00, (2.2)

!4./(i-,>iq«,i<*lMIU, + e-Theorem 2.b.7 of [8] says that each linear bounded operator T: c0 —» /, is 2-abso-

lutely summing, consequently

tr2(C) < KG\\C\\^X (2.3)

where C is the operator defined by the matrix C and KG is a universal constant.

Note now that by Holder's inequality we have

|/7||= suP (ÍI4AI") '<|A-iiftii,<i\i=i /

ai-/') (2.4)

where D: /j —> /p is defined by the matrix D and b = (6,-)," i G /,. Hence

(oo \l/2 / oo \l/2

= (hy(2.2)) = lf1\\DC(ei)\\2/p\l\,= i /

<||ß||(f ||Ce,.||j < (by (2.3) and (2.4))

<IMI,/U-,)-KG||qk.- sup Í2i«,i2)1/2

< (by (2.2))

< KG(K\\A\\X,P + e) < (by (2.1))

»/>

2 a¡X¡i = i

< KKGM + eKG.

Since e is arbitrarily small it follows that, for every y = SJLi aixi G 7,,

(¿N1/2

< MKKn 2 a,-*,li/>

(2.5)



Consequently the operator U: lp —* l2 defined by U(x¡) = e¡, i = 1,2,.

continuous and verifies

is

IMI,,2 = SUP 2 -Mi = i

: a = 2 «/*,■> \\a\\p < 1 [ < KKGM,i=i

(2.6)

Denoting by Sq the unit ball of lq, 0 < q < oo, and by T^S^) the closure for the

topology of /, of the convex and balanced hull of Sp, it is easy to see that

S, £ r(1)(Sp. Then, by (2.6), it follows that

£/(/, n sx) c u(f^(sp) n lp)

Q (KKGM)-lTw(S2) c (KGKM)-XS2,

and this relation implies that U can be extended to an operator V: lx -* l2 such that

we have

|| V\\ < KGKM. (2.7)

But Theorem 2.b.6 of [8] says that each linear bounded operator V: lx —» l2 is a

1-absolutely summing operator and ttx(V) < KG\\ V\\ < KGKM.

Applying Theorem 1.6 it follows that V G Pp(lx, l^, hence there is a positive

constant Mx depending only on p such that

■n (V) < KGKMMX = A\. (2.8)(2.8) implies that

(oo \l//> / oo \i/p

2kr) =(2iin«,*,)if)

< *i sup 2 |a, 2 V»7=1

P\x/P

|X,|<1\,= 1

On the other hand, denoting by C the adjoint of the operator C, we have

,p\ i/>

(2.9)

sup 2 io,r|\|\»//>

2 V,77=1

< (by Holder's inequality)

2 Vy.7=1

•HIp/O-p)

-IMUi-rt-Hn«c.i < (by (2.2)) </C||/l| + e

< (by (2.1)) < KM\ 2 a,*,i=i

\/p+ E. (2.10)

(2.9) and (2.10) imply that, for every y = 2°», a,*, G /

(oo \WP

2 |a,r <KXKM

Thus (*,),. ~ (*,),• D

2 a,*,i=i

Wp

kxkm[ f |a,f)!//>


436 NICOLAE POPA

3. Symmetric basic sequences in d(w,p), 0 <p < 1. In this section we prove the

analogues of Theorem 3 and Lemma 1 [1] for 0 <p < 1. We show moreover the

unicity of the symmetric bases in d(w, p) in the case that this space is not included

in /,. Finally we state some open problems concerning the symmetric basic

sequences in d(w,p), 0 <p < 1.

In the proofs of the following two results the techniques of Altshuler, Cassaza

and Lin [1] work almost unchanged.

Lemma 3.1. Let (xn)^°_, be the canonical basis in d(w,p), 0 <p < 1. If y„ =

2fl*' +1 a¡x¡, n = 1, 2, . . . , is a bounded block basic sequence of (xn)x_x such that

limn an = 0, then there is a subsequence of (yn)^x which is equivalent to the

canonical basis of L.

Proof. Since every change of signs and every permutation of the integers

induces an isometry in d(w, p) we may assume, by switching to a subsequence if

necessary, that (a¡)°^x is a nonincreasing sequence of positive numbers. Moreover

we may assume that ||.yj| = 1 for n G N.

Now let 0 < e < 22(2^ - \yl/p(2p+x - l)~x/p. Using the facts that lim„ a„ = 0

and HjrJIâ, = 1 for every n G N, it is easy to construct by induction two increas-

ing sequences of integers («,)j*l i and (/))_," i such that

Pn><>'j<Pn¡ + x,

Ôo = 0,7

Qj - 2 (/\ + l - Pnk) < 0+1 - TV,' J = l> 2' ■ ■ • ' (3.1)k=\

l/P

2 (".Y-",-,..) <e/2J+i.

For the sequence of scalars (A,-),™ x we have

2 V,7=1

2 a,7=1

2 «,*, + 2 «,*,= />„ + ! / \''=0 + '

oo I Pt+i

2 M 2 <*txi7=1 \'=0+l

2 \ 2 a¡X¡7=1 V'=/>,+ !

Pm+l

2 \\ 2 a,xt7=1 \i-0+l

- 2 IV7-1

2 «iî

> (since {/* - ^ + 1, . . . ,Pak + l - Pnk)

n {rj - p»j + l> ■ ■ ■ -/Vi ~ p",) = 0and

{rk + 1, ...,7\ + i} D {/) + 1.*Vm} - 0 for k*j)



> 2 fcfl 2 («,r--Q.,_0+,. - 2 (e/^ri^r7=1 \'=o+l / 7=1

> 2 |a/( 2+' («,r--,.-J -\ei->iv- ir']-8up|^7=1 \' = r,+ l 7 7

> 2 |A/[l - ep2-p(i+X) - t¡n^(lf - I)"1]7=1

> 1 -ep(2p+x - 1)

22'(2'' - 1)

On the other hand we have

2 iV-7-1

2 V*7=1

< 2 IM>Ji = 2 \h\p- a7=1 7=1

We shall use forward a special block basic sequence. Let (•*„)"_ 1 be a symmetric

basis in the /7-Banach space X. If 0 =?*= a = 2~_i a,,*,, G A' and if (/»,-),*! is an

increasing sequence of integers, let >>â) = 2^ +1 a¡_p x¡, n G N. Then (y„a))n°„x is

a bounded block basic sequence of (xn)"_,, and we shall call it a block of type I 0/

(*X=,-

Theorem 3.2. Every bounded block basic sequence of (xn)n°^x in d(w,p), 0 <p <

1, has a subsequence equivalent either to the unit vector basis of lp or to a block basic

sequence of type I of (*„))■= i-

Proof. Let yn = 2^ + , a,*,, n = 1, 2, .... We may assume that H^JI,,*, = 1

and that ap +x > ■ ■ ■ > ap > 0 for every n G N. If supn(pa+l — pn) < + 00, then

it is clear that (y„)n ~ (*„)„, hence (.y„)j¡°_ ! is equivalent to a block basic sequence

of type I of (xn)x-v Assume now that sxxr>n(pn+x - pn) = 00. Let b¡ = sup„|aPn + ,|

for 1* G N. It is easy to prove that lim, b¡ = 0.

Case I. Assume that for every e > 0 there exists m such that \\\ZP"JP +m a¡x¡\\ <

e for every n so that/7„+1 — />„ > m. Since supn(//n+1 — /?„) = +00, we may assume

that pn + 2- pn+x > Pn+X- Pn for every n G N. Define now z„ = 2fi+1'";'" «/+,„-*,•>

« G N. Then ||zn||PjM, = H.yJL^H, = 1 for n G N. By hypothesis and using the fact

that there is a subsequence (nk)^x such that the sequences (aj+p )"_, converge

simultaneously for i < m, we can find a Cauchy subsequence of (z„)*_,. Hence we

may assume that limn zn = z = 2°L] c,*, G d(w,p). It is clear that z =£ 0.

Since (y„)%L ! is a bounded block basic sequence of (.*„)£_, it is well known that

K = supJI-PJI < +00, where 7>n(2°°_i a¡y¡) = 2"_, aiyi (see III.2.11 of [10]). Conse-

quently we can find a subsequence (z )*

Define now

such that 2" < 1/2 JC.

p^+i

Ui = Zd Ck-pnXkik=P« + l

i = L 2, . .


438 NICOLAE POPA

Then (w,-)," , is a block basic sequence of type I of (xn)na_x and 2°l1||>'n,. — u¡\\pw <

2/liH^ - z\\PiW < 1/2K, hence by the Krein-Milman-Rutman Theorem (see The-

orem III.2.13 of [10]) it follows that (y„)^x ~ («,),_,.

Case II. There exists an e > 0 such that for every m G N there exists n(m) such

that pn+x — p„ > m and ||2f^¡ + m a¡x¡\\PtW > e. Then there exists an increasing

sequence of integers (i,-),™ i such that

Pn,*\

t>n,+ i-Pn,>i and 27 =/>* + '

> e for some i G N.

P.w

Since lim, b¡ = 0 we may assume that (a)JL \ is a decreasing sequence. Let

Pm+l

z, = 2 ajXj for i G N.7=/%+l

Then e < \\z¡\\p„ < H^H^ = 1 for i G N, also (z,)°l, is a bounded block basic

sequence of (*„)"_! and the coefficients of z, converge to zero. By Lemma 3.1 it

follows that there exists a subsequence (r,),"] of (z,)°l, which is equivalent to the

canonical basis (<?,.)£, of lp. Since (y„)T-\ dominates (ti)xmX (i.e. ||2r_, ¿yjl,,«

< + oo implies that ||2Jli 6,/|l;>>w < oo for every sequence (è,-),",) then

l|2J_! ¿¿yJI^H. < °o implies that '2'*Lx\di\p < oo for every sequence of scalars

(</,-),",. On the other hand, since Wy^]^ = 1 for every i G N, 2£,|4|/' < oo

implies that ||2f-. 4^11,,* < «. hence (yj, ~ (e,)t. DIn the remainder of this section we study the unicity of the symmetric bases of

the subspaces of d(w,p), 0 <p < I. We shall often use the following notion. Let X

be a p -Banach space with a separating dual X (this is the case whenever X has a

basis). We consider on X the finest locally convex topology weaker than the

original one i.e. the Mackey topology on X. It is easy to see that the Mackey

topology on X is generated by the neighbourhoods ((l/n)co(S))"_,, where S =

{x G X: \\x\\ < 1}. Then the completion of X in the Mackey topology, X, is a

Banach space. It is interesting that d(w,p), 0 0 there is n G N such that

oo

sup 2 |«U./-W< <e (3.2)v i = n

uniformly for a = (a,)°lx £\ A. (The supremum is taken over all permutations of

integers.)

Proposition 3.4. Let 0 <p < 1. Then d(w,p) = /, if and only if d(w,p) c lx.

Moreover if d(w,p) rj: /, then d(w,p) 96 /, (i.e. d(w,p) is not linearly homeomorphic

to /,).

Proof. If d(w,p) d; /,, let a = (ai)xmX G d(w,p) \ /„ A = {x¡: i G N} and / G

d(w,p)'. We shall show that

lim /(*,.) = 0. (3.3)



Indeed if (3.3) is not true, there exist the sequence (/*)"_ i of integers and a > 0

such that \f(x¡ )\ > a for every k G N. We consider b = (b¡)fix, where

_ j aj sign fixh), i = ij,j G N,

0, i ¥* ij,j G N.

It is clear that b G d(w,p) \ /,. But oo = a. 2°l,|a,| < |2°l,/(x,)è,| < oo, which is

contradictory. Then (3.3) is true and A is weakly relatively compact. On the other

hand (3.2) is not verified for A and p = 1, hence A is not a relatively compact

subset of d(w, 1). Since the canonical mapping i: d(w,p)—> d(w, 1), 0 <p < 1, is

clearly continuous, it is obvious that on d(w, p) the topology induced by d(w, p) is

stronger than that induced by d(w, 1), consequently A is not a relatively compact

subset of d(w, p).

If d(w, p) »A, then, A being weakly relatively compact in d(w, p) (since clearly

d(w,p)' = d(w,p)') it is also relatively compact in d(w, p), which is a contradiction.

Thus d(w,p)?é /,. Conversely, if d(w,p) c lx, we denote by /: d(w, p) —* /, the

canonical mapping. We shall show that / is continuous, that is there exists K > 0

such that, for every a G d(w, p), we have

2 H < *|M|#. (3.4)i=i

It is clear that it suffices to prove (3.4) only for positive decreasing sequences

(a,),°l, G d(w,p). If (3.4) is not true, then for every n G N there is the positive

decreasing sequence a{n) = (a/*0),", G d(w, p), such that

2"||a(n)|^ < 2 a\n\ (3.5)i = i

Denote by £<"> = l-lé%jL*¿». Then ||2-., &<">|L„ < 2r_1||Z><%„ <2"_11/2"P < oo, consequently it follows that b = 2"_, b(n) G /,. But (3.5) implies

that 1 < 2°°_i bfn) for every n G N, hence ||2~_1¿>(,,)||1 = 2~_, 2°°_i bfn) = + oo,

which is a contradiction. Hence / is continuous. On the other hand the canonical

mapping J: I ^>d(w,p) is continuous and consequently the extensions /: lp —»

d(w, p) and J: d(w, p) -» /, are continuous. Since clearly lp = /, for 0 1.)

Example 3.5. Let wn = (1 + |log «|)_1 for n > 1 and 0 <p < 1. Then d(w,p) c

Hi-

proof. Our proof is indirect. We denote by

M(t) =

tpG(0, 1],

1 + |log i| '

[0, t = 0.

Then M(t) is a continuous nondecreasing function on [0, 1], with

sup(fE(0 X]M(2t)/M(t) < oo, that is an Orlicz function (see [5]). We consider now


440 NICOLAE POPA

the locally convex Orlicz sequence space lM = {a = (a,)*,: \\a\\M = 2°1, M(\a¡\)

< + a}. We shall show that

lM = d(w,p). (3.6)

It is clear that lM = lM , and that d(w, 1) = lM¡ implies that d(w,p) = lM, where

f

Mq(t) =

f6(0,l],1 + |log t"\ '

0 <q <+oo.

0, t = 0,

Consequently it is sufficient to show that

Theorem 4.e.2 of [8] says that d(w, 1) = lM if and only if there exists y > 0 such

inat

2 l/W'\ywn)<+oo (3.7)n = l

where W(x) = (I + log x)-1 for x > 1.

In our case W~x(ywn) = ex/^y~i)nx/y, hence, for every 0 < y < 1, (3.7) and also

(3.6) are true. But Theorem 3.3 of [5] shows us that d(w,p) = l^, where M is the

largest Orlicz convex function on [0, 1] such that M(x) < M(x), x G [0, 1]. Since

K(p)t < tp/(l + |log i|) < tp for / G (0, 1], where K(p) > 0 depends only on p, it

follows that M is equivalent to the function N(t) = t, hence

dJ^Tp) = lû = /,- (3.8)

By Proposition 3.4 it follows that d(w, p) c /,. □

Remark 3.6.1. Since M(t) is not equivalent to N(t) = tp (i.e. there are not the

relations 0 < inf,e(0 X]M(t)/N(t) < sup/e(0 x]M(t)/N(t) < + oo) the previous space

d(w,p) is a/7-Banach space, other than lp, whose dual is lx.

2. This space is moreover an example of a /7-Banach space X with a unique

unconditional basis, other than lp, for which X has a unique unconditional basis.

Indeed limx_>0 M(x)/x = +oo, hence by Theorem 7.6 of [5], d(w,p) = lM has a

unique unconditional basis. □

There are spaces d(w,p), 0 0 and

0 < y < 77/(1 - p) (where 0 / e N- (3-9)

nj+x-nj<bj\ j£\N. (3.10)

wn < l/j ifn >«,,/GN. (3.11)

Thend(w,p)r\_lx.



Proof. Let

if i < nx,

" \j-ß/p(nJ+x - nj)-l/p Un, < / < «,+ 1,

where

0<ß<p-(l-p)y. (3.12)

By (3.9) it follows that (a,)* ] is decreasing to zero and (3.11) implies that

"7+i w

\\a\\P,w = 2 wt + 2 2,_i j=x , = „.+ ! jß(nJ+x - nj)

"l OO 1

< 2 y>i+ 2 -^z < + ».1=1 7=1 /

On the other hand

"j OO B7+l

HI. = 2i+2 2 jß/p(„ _ „ v/p

= «,+ 2

, = 1 y.i , = „, + 1 jP/P(nj+l- „.)

1

, = > /"'(n,+I - -,)(,/'>

CO

> (by (3.10)) > nx + bx~x/p 2, ,£/>. ,-Y<l//'-l)

7=1 V 7

oo ,

> (by (3.12)) > », + ô1-1» 2 - = oo,7=1 /

thus a = (a,),°l, G /,. D

The space d(w,p), 0 1, satisfies the conditions

of Proposition 3.7. In Theorem 4 of [1] it is proved that every two symmetric

bounded bases of a subspace of d(w,p),p > 1, are equivalent. Similarly we can

state the following still open problem.

Problem 1. Let X c d(w,p), 0 „)"_ i and (z„)~_, equivalent?

We are unable to give an answer to Problem 1, however the following theorem is

true:

Theorem 3.8. Let d(w,p), 0 <p < I, such that d(w,p) (£ /,. Then d(w,p) has a

unique bounded symmetric basis.

Proof. Let (y„)^x be a bounded symmetric basis of lp other than (x„)~_,. By

Proposition 3.4 the hypothesis implies that d(w, p)?é lx, hence

lim x'„(ym) = 0 for every n G N, (3.13)tfl

where (x'n)n"=x is the biorthogonal sequence in d(w,p)' associated to (xn)n°_x (i.e.

x'n(xm) = dm„ for every m,n£\ N). Indeed if (3.13) is not true for n0 G N, there

exist the subsequence (ym)°tx of (y„)x=x and a > 0 such that a < \x'n (>»_)| for

every i G N. Then for every scalar (a¡)?=x, there are e, = ± 1, / = 1,2, ... ,n and


442 NICOLAE POPA

0 < M < oo, such that

a 2 Kl < x,ni 2 wJ a¡e¡y»

2i=i IM <^2i=iwhere ||x||~h, is the norm of the element x G d(w,p). This inequality shows us that

d(w,p) ä! /,, which is a contradiction. (3.13) implies by Proposition 3.1 of [5] that

there exists a subsequence (yn)°Lx of (y„)™„x which is equivalent to a block basic

sequence of (.x„)^L i-

The basis (>»„)"_ x is a symmetric basis, then (y„)fL \ ~ (y„)™= i and consequently

we may assume that^ = 2?™*' +x b¡x¡, m G N. Moreover by Lemma 3.1 it follows

that lim supjaj * 0. (If not then (yn)n ~ (e„)n.)

Since (x„)£L, is symmetric, we may also assume that there is e > 0 such that for

every m G N there ispm + 1 < km < pm+x so that ¡¿/¿J > e. If 2"_, a^y„ converges

in d(w, p), then for every n G N,

1< — M**,

1

p,w

<v<

hence 2"_i a„JC„ converges in d(w,p). If we interchange the roles of (*„)*_, and

(y„)™= i we deduce the equivalence of these two bases. □

We can state a weaker version of Problem 1.

Problem la. Let d(w,p) c /,, 0 <p < 1. Is the canonical basis of d(w, p) the

unique bounded symmetric basis of d(w, p)l Finally we have

Theorem 3.9. Let X c d(w,p), 0 <p < 1, be a subspace which has a bounded

symmetric basis (yn)n°-x verifying the equality (3.13). Then any other basis (z„)"_, of

X, which has the same properties as (yn)„°= x is equivalent to this.

Proof. If X œ lp, 0 ", = y«.+i■=«, ■..a,*,, n = 1, 2, ... , and (zn)"_, is equiva-

lent to a block basic sequence vm = 2^;rm + i b„yn, m = 1,2, ... , of (yn)„°. More-

over we may assume that lim sup,,.^ bn =£ 0 (otherwise Lemma 3.1 implies that

(z„)„ — (e„)„ which contradicts our assumption). Reasoning as in Theorem 3.8 we

obtain that (z„)"-i domainates (y„)n°^x. By interchanging the roles of (>»„)"-1 an<^

(zn)^°_, we deduce the conclusion. □

Let us mention the following problem.

Problem 2. Is there a subspace X c d(w,p), 0 <p < 1, different from ¡p and

d(w, p), such that X = /,?

Remark that a negative answer to Problem 2 and a positive one to Problem la

imply that Theorem 3.9 is true without any restriction concerning the basis (^„)*-i-

4. Complemented subspaces and sublattices of d(w,p), 0 < p < I. We prove now

the version for 0 <p < 1 of Theorem 1 of [1]. By (x„)™-i we mean forward the

canonical basis of d(w, p).



Lemma 4.1. For every bounded block basic sequence (yX~\ °f (xn)n = i there is a

block basic sequence of (yX-1 which is equivalent to the canonical basis of I.

Proof. Let yn = 2f=¿ +1 a¡x¡, n = 1, 2, ... . Let us remark that, since

íujJI.t'JUh. > 0, 2°li a¡xi does not converge in d(w,p). On the other hand, if

sup„||2"=) a¡x¡\\pw < oo, then 2°_! a¡x¡ converges in d(w, p). Hence

suPk<m\\2"=kyi\\p,w = + °o.

Let (pX= ! be an increasing sequence of integers such that

sup 2 ä•Pn+i

= 00.

Also let

Pn+\

2 y>'=Pn+ 1

l/P

2 y,i=p„+ 1

« G N.

Then the block basic sequence (zX= i or (.On3-1 satisfies the conditions of Lemma

3.1 and, consequently, there is a subsequence (zn)°l, of (z„)"=1, which is equivalent

toOX-i- □Using Lemma 4.1 we can prove

Theorem 4.2. Let X c d(w,p), 0 <p < 1, be a (closed) subspace of infinite

dimension. Then there is a closed subspace Y G X such that Y m i'.

Proof. By Proposition III.2.15 of [10] it follows that X contains a bounded basic

sequence (yX-i which is equivalent to a block basic sequence (zj*., of (xX-v

Then Lemma 4.1 gives us a subspace of Sp{z„: n G N) which is linearly homeo-

morphic to lp. Consequently X contains a subspace Y « lp. □

Remark 4.3. In Corollary 17 of [3] it is shown that every (closed) subspace of

infinite dimension X of d(w,p), 1 < p < oo, contains a subspace Y complemented

in d(w,p) and linearly homeomorphic to lp.

This assertion is not true in the case 0 /2y2+x e¡, n = 1,2,. . . , does not contain any infinite dimensional

subspace which is complemented in lp. Consequently, let A' be a subspace of d(w, p)

linearly homeomorphic to lp and let (z,),°l, be a bounded basis of X. We consider

the block basic sequence u„ = n~x/pHf2^n-2\)/2+\ zr n = •> 2,- Then Y

= Sp{«„: n G N} « lp does not contain any complemented infinite dimensional

subspace of d(w, p). □

Remark 4.3 shows us that there are subspaces linearly homeomorphic to lp (in

fact isometric to lp) which are not complemented in d(w,p) for 0 <p < 1. We can

prove moreover that there are examples of spaces d(w,p), 0 <p < 1, without any

complemented subspace linearly homeomorphic to lp.

Example 4.4. Let wn = (1 + |log n\)~x for every n G N and let 0 <p < 1. Then

any complemented subspace of d(w,p) with an unconditional basis is linearly

homeomorphic to d(w,p) çé lp.


444 N1COLAE POPA

Proof. By Example 3.5 and Remark 3.6 it follows that d(w, p)lM sé lp, where

M(t) =

t

0 if t = 0.

Moreover, the canonical basis of d(w, p) is the unique unconditional bounded basis

of d(w, p). But a theorem of Kalton (see Theorem 7.2 of [5]) shows us that if lF is a

nonlocally convex Orlicz sequence space, then any two unconditional bounded

bases of lF are equivalent if and only if any complemented subspace with an

unconditional basis is isomorphic to lF. We conclude applying this result. □

Example 4.4 motivates the following question:

Problem 3. Let d(w,p), 0 <p < 1. It is true that there does not exist a comple-

mented subspace of d(w, p) which is linearly homeomorphic to /p?

First we give some results which seem to indicate an affirmative answer to

Problem 3. First we study the positive complemented sublattices of d(w,p),

0 || for every a, b G X. (4.1)

We call such a space X a p-Banach lattice. (Analogously it is defined a Banach

lattice.) It is clear that, extending the order relation to X (whenever the last space

exists), A' is a sublattice of X. Let A" be a vector lattice and Y c X a sublattice of X

(i.e. x G Y implies that \x\ G Y) which has the property that x G X, y G Y and

|jc| < I.y I imply that x G Y. Then we call Y to be an order ideal of X. If A c X is a

subset of X, then we denote by IX(A) = {x G X: 3a G A and 0 < X G R, such

that |jc| < X\a\}, the order ideal generated by A.

It is clear now that d(w,p), 0 <p < 1, with the canonical order relation is a

/7-Banach lattice and, moreover, an order ideal of the space co of all sequences of

real numbers. Now we can extend (under certain conditions) Lemma 2.a. 11 of [8]

to the/7-Banach lattices.

Lemma 4.5. Let X be a p-Banach lattice which is an order ideal of u>, (vX-1 G co

a sequence of positive real numbers, 0 < yn = 2,<=0 a¡x¡, and 0 < zn = 2ye+ BjX„

n G N, two bounded basic sequences of X (where xn = (dn¡)fLx for n G N) such that

°n n »rm = 0 for every n, m G N. If there exists a positive and continuous projection

P from X onto Sn{vnyn + zn: n G N}, then the sequence (zn)n dominates (v„yn)n.

Proof. Let P(y¡) = 2°°_, c/)(W + zj) and 7>(z,) = 2°1, dfi\^r + z¡), i =

1,2,..., where cj¡) > 0 and ¿W > 0. Since the basis (xX-1 is clearly an uncondi-

tional basis of X, there is a positive and continuous projection Q such that

Q(xn) = xn for n G a„ where / G N, and Q(xn) = 0 otherwise. Then QP(z¡) =

2J1, dJ-'^Vjyj, i G N, and g/1 may be considered as an operator from Sp{z,: i G N}

to Sp{j>,,: n G N) defined by the infinite matrix with positive entries (<j^y)5-i*

Then the diagonal matrix defines an operator

7): Sp"{z,: » G N) -► /„( Sp~{>>,.: / G N}) = /-(Sft.*: i G N}).



D is obviously positive and 0 < D(x) < QP(x) for every 0 < x G Sp{_y,: i G N}.

Since X is a/7-Banach lattice we have

\\Dx\\ < Hß/'xll < Af||x|| for every 0 < x G Sp{.y,: i G N},

consequently 7) is a continuous operator.

Assume now that 2„°_1 a„z„ converges in Sp{z,: i G N). Then Z)(2"_, a„zn) =

2?-i «„^V^ converges in fjr(^{^: Í G N}). Since 0 < c„")(V„ + z„) < P(yn)

for every « G N, then |c„n)| < M, for n G N. On the other hand vncnn) + d^n) = 1,

n = 1, 2, ... , hence lim„ i7„(n) = 1 and, consequently, 2°f_, anvnyn converges. □

We can now prove

Theorem 4.6. Let X be an order ideal ofu, which is a p-Banach lattice and assume

that its canonical basis (*,,)"_ i 's symmetric. If (yX-1 's a positive block of type I of

(xX=i then (yn)n ~ (xn)n '/ and only if E =Sp{y„: n G N} is a positive comple-

mented sublattice of X (i.e. there exists a positive and continuous projection P from X

onto E).

Proof. Let 2„°_, a„xn G X, a, ¥= 0 and yn = 2fb¿+i <%-£*/ for every n G N,

where a¡ > 0. We may assume that all a¡ < 1. Since (xX-1 is a symmetric basis of

X, there exists M > 1 such that

J_M 2 KXP.+ \ 2 M»

n=l

< m 2 *«j^+j=i p.+i

for every 2„ Ä„x„ G X. Suppose now that (y„)„ ~ (x„)„- Let 7C > 0 such that

II2"-i*wJI <A"H2r-i^JI for every 2„ bnxn G X. Define PÇZ„°_X bnxn) =

2?-.(*,„+./«.K for 2„ bnx„ G A". Since (>-„)„ - (x„)n, P is well defined, 0 < P

and || P || < ATa-''.

Conversely, let P: K -^ E be a positive and continuous projection. If x =

2„ è„xn G A' and ||jc|| < 1, we choose the sequence of integers 1 = «, < n2 < . . .

such that ||2°°=n( bjXj\\ < 1/2', i = 1, 2,

put

For n¡ < m < n, + 1, / = 1, 2, . .

2 «rô7=1

Pm +7 if/>m + ' <Pm+V

tipm + i >7>m+.,

we

and

(ym - zm)/\\ym - 'mí ¿y», + *-

o iiym = z„

Let vn = ||.y„ - z„|| for n G N. Then i>m, *-., zm are positive, ym = »<mw>m + zm for

m G N, (Ob-i g co an<i inf(z«» wm) = 0 for m,n£\ N. Since 0 < a, < 1, / =

1, 2, ..., it follows that


446 NICOLAE POPA

2 bmznn = \

2 2 bmZn, = 1 m = n,

OO "i+l-l Pm+¡-Pm

2u 2-¡ "m 2-i XP„+j7=1,= 1 m = n¡

< (since (tc,)°°_ , is a symmetric basis) < M

oo ";+l"i*i- 1

2 2 bmXpm + l, = 1 m - n,

< A/2i=l

2 */v'i7 = ",

1

<m22 2 V/ ■

< A/22 — <+oo.,= i2'

Consequently 2„°_1 £„z„ converges and, applying Lemma 4.5, 2*.! bHPnwH con-

verges, therefore ||2"=i b„y„\\ < oo. Conversely if 2"_! bnyn converges, then

2"_] axbnx +1 converges. Since (xX~\ is a symmetric basis, then a,(2°f_1 bnxn)

converges. Consequently (yn)n ~ (xn)n. D

Proposition 4.7. Let X be a separable p-Banach lattice with the property that

every order interval [0, x], where X 3 x > 0, is compact. Then there is in X a

normalized sequence (xX=\ of positive pairwise disjoint (i.e. inf(x,, x) = jc, A x¡ =

0, i ¥=j) elements such that (xX-\ & a basis simultaneously of X and of X.

Particularly (xX=\ /J an unconditional basis of X. Moreover X is an order ideal of X.

Proof. We prove first the second assertion. Let y G X and z G X such that

0<z<y.lfO<zn£\X and lim„ zn = z in X, then, since X is a/7-Banach lattice,

limn zn/\y = z/\y = z in X. On the other hand zn /\ y £\[0,y] <z X and, by

hypothesis, there is a subsequence (z„ A y)k°= i which converges in X (and conse-

quently in X). Hence lim„ z^ f\ y = z G X and X is an order ideal in X.

Remark now that the hypothesis implies that [0,y] is compact in X for 0 < y G

X. But a Walsh's result (see [13]) asserts that a Banach lattice X such that every

order interval [0, x] is compact, has a normalized basis of positive pairwise disjoint

elements. Consequently there exists a subsequence (xX-1 OÍ X of positive pairwise

disjoint elements, with 0 < inf„||xj|~ < sup„||.xj|~ < oo (where ||tc||~ is the norm

of the element x £\ X), such that we have a unique expansion x = 2^_! a„x„ with

0 < an, for every 0 < x G X.

Since A" is an order ideal of X, there is a subset A Q N such that x, G A" for

every i £\ A. Let 0 < x G A". Since [0, x] is compact in X, it follows that x =

H„eA anxn, the convergence being with respect to the topology of X. Hence

(xn)n(EA is a basis of A". It is easy to see that (xn)nSA is a bounded basis (see also

Proposition 3.2 of [5]). Consequently we may assume that \\xn\\ = 1 for every

n G A. Since X is dense in X and x¡ /\x¡ = 0 for / ¥=j, it follows that A = N. The

remaining assertion is a consequence of Lemma 1.1. □

Remark now that in view of Proposition 3.3 it follows that [0, x] is a compact set

of d(w,p), 0 <p < 1, for every 0 < x G d(w,p). Consequently, by Proposition 4.7,



every sublattice of d(w,p) has a basis of positive pairwise disjoint elements.

We can now state the analogue of Corollary 14 of [31.

Theorem 4.8. Let E be a positive complemented sublattice of d(w, p), 0 <p < 1,

which has a symmetric basis. Then E is linearly homeomorphic either to d(w, p) or to

'p-

Proof. Let (yX= i be the symmetric basis of E whose elements are all positive

and pairwise disjoint. If _y„ = 2°1) tnix¡ for n G N, then limn tM = 0 for every i G N

and, by Proposition 3.1 of [5], there is a subsequence (yn )£_, of (yX-i which is

equivalent to a bounded block basic sequence (zX-i ot (x„)T-\- Moreover

repeating the proof of Proposition 1 .a.9 of [8], it follows that we can choose z„ > 0

for n G N such that Sp{z„: n G N} is a positive complemented sublattice of

d(w,p). (yX-i being a symmetric basis it follows that (y„)„ —(zn)„. Then we can

apply Theorem 3.2, consequently if E sé lp, there exists a positive block of type I

("/.)*"» i OI (xn)«°= i sucn that (y„)„ — ("„)„• Using again the proof of Proposition

l.a.9 of [8], we may assume moreover that Sp{w„: n G N) is a positive comple-

mented sublattice of d(w,p). By Theorem 4.6, we obtain now that («„)«-1> a'so and

(yX- v is equivalent to (xX-. • D

In connection with Theorem 4.8 we can state a weaker version of Problem 3.

Problem 3a. Let d(w,p), 0 <p < 1. Is there a positive complemented sublattice

E c d(w, p) linearly homeomorphic to lp1

If we have a negative answer to Problem 3a, then Theorem 4.8 shows us that the

positive complemented sublattices with a symmetric basis of d(w,p), 0 E is a contraction, i.e. ||P|| < 1.

5. Positive and contractive complemented sublattices of d(w, l/k), k G N. In this

section we shall characterise, under certain conditions, the positive and contractive

complemented sublattices of d(w, l/k), k G N. We prove first an inequality:

Lemma 5.1. Let k G N, (w,),°°., G c0\ /, such that wx > w2 > ■ ■ ■ > 0 and a,

= • • • = a, > a/+1 > a/+2 > • • • > am > 0 for some I < m. Then denoting by

sn = E;=1tv,fy

c(i) = (¿)(«,+ ,)1A(«/-1A - «iV.'^hw,*-1

and by

Í 1 for n > I + I,

10 forn<l + I,

we have the following inequality

K = 2 *,*(«, - «, + ,) + snka„ + d,(n)C(l)

(I, "''*< 2 «,- -w, = Bn for every n < m. (5.1)


448 NICOLAE POPA

Proof. We use the induction for n. If n < / it is nothing to prove. Now let/ i 1 TUn = I + I. Then

a!+x = a,,* + (£, - S*)«,+I + (i )«■/*(«;--a _ a.+-.A)vW/*-.

= «a* + «/..((^"/v^-2 + • • • +<.) + (lyxw-^-^sr1

< (since o/+, < a,,/ = 1, . . . , /)

+ ((ikw"^'"?^*- • • +«/.,<.)VV /c/ /

(i; «,-,)'

/+i \*

(2 «,IA--,|

Thus (5.1) is proved in this case.

Assume now that (5.1) is true for n — 1 > I + 1. We have

A -4,-1 +(í* - i*-lK

< (since a, > a,+, for /' < n)

(DC?:«'<^-.+ *-i•w. «y*-^n + • • • +«1.^n n n n

< (by induction hypothesis)

< s «,"*■», +(¿)( % ,■"-,)">• ,i/*

= (¿ «T •-,)"= *,. D

Lemma 5.2. If/r/i /Ae notations of Lemma 5.1, if (ßi)T-\ are positive numbers such

that

2 A- < ** f°r every n < m,

then

1-1

2«,A + 3/(")c(/)<(2«/A->v,) /or n < m.

(5.2)

(5.3)



Proof. If n < m, we have

2 «,A + d,(n)C(l) = «„( 2 a) + («»-. - «j("¿' a)

+ • - - + (a, - «j)A + 9/(")C(/)

< (by (5.2))

< <v„* + («.-i - Otf-i + •••+(«,- aj)if + 3,(»)C(/)

<(by(5.1))<(2«,,A-^) • D

Using Lemma 5.2 we can prove now

Proposition 5.3. If p = 1/A:, 1 < k G N, a«<7 Tf ¿s a positive and contractive

complemented sublattice of d(w, p), then there is a sequence of finite pairwise disjoint

subsets (sX=i of N such that un = (2""_,w,)"*(2,e^ X¡) for n G N, constitute a

normalized basis of E, dn being the cardinal number of a„.

Proof. By Proposition 4.7, E has a normalised basis (uX-\ OI positive pairwise

disjoint elements of E. Let un — 2,eo otinx¿, where (oX-\ is a sequence of pairwise

disjoint subsets of N and a,„ > 0 for / G a„, n G N.

If dn = + oo, then we write a„ = {ty: /' G N}, and, since (uX-\ is normalised, it

follows that (ain)°Lx G d(w,p). Consequently there exists a permutation of integers

ir„ such that 7/n(a„) = on, (ct^ (i)X= i is a decreasing sequence, and trn(i) = /' for

every / G o„. If an < oo, such a permutation mn exists obviously. Since the norm

|| || is invariant under permutations of integers, we can find an isometry

T: d(w,p)^d(w,p) such that T(uJ= tm, where tm = 2^., a„m(ij)m for every

m G N. Then Q = TPT~X: d(w,p) —»Sp{im: m G N) is a positive and contractive

projection whenever P is a positive and contractive projection onto E. Conse-

quently, we may assume that the coefficients (a,m),eo of um are decreasingly

ordered for every m G N. Since P is a positive projection, we have

/>(*,.) = ßinun where ßm > 0 for every / G an, n G N, (5.4)

and moreover

(CO \*

2 (<v)'x* • wj for evefyrt e N- (5-5)7=1 /

By (5.4) and using the fact that IIPII < 1 we have

<sup2|Y.(,)rA-w,. (5.6)2 ( 2 yAIh,n = 1 V i e a„ / 1=1

for every element (y,)fl, G d(w,p). Consequently, taking y¡ = y¡ = • • ■ = y, =

1 and y, = 0 fory =£ /'„... , im, where m < än, we obtain

m

2 /V<*¿ forA«<rÍ„,«GN. (5.7)7=1


450 NICOLAE POPA

If an = oo, then, since lim, ain = 0, there is an integer / G N such that a¡¡n > o, „

and a, „ = a, „ for/' < / — 1.

By (5.3) we have

2 a,nA„ + C(l) <7=1 (!<»,)'

for every m > I + 1.

Now, by passing to the limit over m —» oo in the previous inequality, we obtain, in

view of (5.5).

OO / 00 \*

i = 2 v8*»+ c<'> 0, this is a contradiction, hence

ön < oo for every n G N. (5.8)

Then, reasoning as above, we get that ain = ajn for every i,j G a, n G N, conse-

quently a,„ = (2°"_! w,)-* for every r G an, n G N. □

We study now the existence of positive and contractive projections onto a one

dimensional sublattice of d(w,p),p = l/k.

Proposition 5.4.(a) Let p = l/k, 1 < k G N, and u = 2,eo a,*,, where \\u\\pw

= 1 a«r/ a, > 0 /or every i G a. 77iert //tere existe a positive and contractive

projection P: d(w, p) -»Sp{«} if and only if the following conditions are satisfied:

a = n, a, = • • ■ = a, = (2"=1 w¡)~k, and there exist the positive numbers /?,

> • • ■ > 0 such that,k

= sk for m<n, (5.9)r« / m \*

a/zi7

2 A = tf. (5.10)i=i

Then P(2r_! «,*,) = (2,go aßjufor every 2°°-, «,*, G ¿O*,/»).

(b) // w = s~k(Z"_x x,) ant/ /„ = (l/n)sk < tm for every I < m < n, then the

operator defined by

It, a'*H(.t. "M*'¿y a positive and contractive projection onto Sp{u}.

(c) If wx < tn for every n > 1 and u is as at the point (a),jhen there is a positive

and contractive projection P: d(w, p) —» Sp{ u) if and only if a = 1.

Proof, (a) If P~Ld(w,p) -> Sp{u} is a positive and contractive projection then, by

Proposition 5.3, b~ = n < oo and a, = • • • = <*,•_ = s~k. Denote now by (/?;)/= i

the coefficients of (P(x¡))"_, decreasingly ordered. Then

2 ßj-7=1

2 A)«7=1

=H2 *»)Ve« '

2 x¡je H

= sk,


LORENTZ sequence spaces 451

where H G a with 77 = m < n. If m = n, then

2 A = p[ 2 xA = 2 *47 = 1 / p,w 7=1

-«*.

Consequently all the conditions are satisfied and obviously P(2°l] a,x,.) =

(2,6o a,-A)« for every 2°°=i «,*, S ¿(w,/?).

Conversely, by (5.3), 2?_, a, A < (2?-i aV* • w,.)* for every a, > a2 > • • ■ > 0,

and, consequently, |2"_, a,A| < supw(2"_! \avii)\l/kwiyk for all scalars (<*,.)?_,. Thus

the operator defined by P(Zf=x a,x,) = (21(Ea a,A)M is well defined, ||P|| < 1 and

P > 0. Since P(u) = s~k(Z"=x ß,)u = m (by (5.10)), it follows moreover that P is a

projection.

(b) If tn < im for every m < n, then, putting ßx = • • • = ß„ = tn, the relations

(5.9) and (5.10) are verified, consequently P(2°l, a,.*,.) = (l//i)(2"_, a,)(2"_, x,) is

a positive and contractive projection.

(c) If there exists P: d(w,p) —» Sp{w} a positive and contractive projection and if

ä > 1, then by (a) it follows that (l/n)sk = (l//t)2"_, A < ß\ < w\ for « > 1,

which is a contradiction. □

We can now state the main result of this section:

Theorem 5.5. If p = l/k, 1 < k G N, and wf < (l/rt)(2"=1 wf for every n > 1,

í/ien the positive and contractive complemented sublattices E of d(w, p) coincide with

the (closed) order ideals of d(w,p). In particular any positive and contractive

complemented sublattice of d(w, p) cannot be linearly homeomorphic to lp.

Proof. If £ is a closed order ideal of d(w,p), then E =Sp{x,: /' G A c N},

hence there is a positive and contractive projection P: d(w, p) —* E. Conversely, if

£ is a positive and contractive complemented sublattice of d(w, p), by Proposition

5.3, it follows that E =Sp{wn: n G N), where un = 2,eo a,x. > 0 for n G N and

(aX= i is a sequence of finite pairwise disjoint subsets of N.

By Proposition 5.4(c) it follows that dn = 1 for every n G N. Consequently E is a

closed ideal of d(w, p).

The second assertion is an obvious corollary of the first. □

The conditions of Theorem 5.5 are verified for example by the spaces d(w,p),

p = l/k, 1 < k G N, with w, = l/¡°, 0 < a < \. Indeed,

2 ", > f"+1 —a =7-^— [0 +»)'-"- 1] >«,/* for every n> 1.,=i 7, x 1 — a

d(w, l/k), where 1 < k G N and w„ = l/na for 1 > a > 1 - 1/rc and « G N, are

examples for spaces d(w, p) for which there is a one dimensional positive and

contractive complemented sublattice which is not an order ideal of d(w, p).

More precisely, for a fixed m G N, there are n > m and a positive and contrac-

tive projection onto Sp{w}, where u = (2"_, w,)_1(2"=1 x,). Indeed, using the

notations of Lemma 5.4(b) we have

1 / r" dx\k „*(i-«> 1r_ < — I I —- I =-;-<-;- for every n G N.

«Wo x") (\-a)k-n (I - a)k ■ nx~k+ak


452 nicholae popa

Consequently, for 0 < e < | min,</<m /„ there is a lowest n > m with tn < e, hence

tn < tj for every 1 < j < n. We conclude applying Proposition 5.4(b). □

Theorem 5.5 suggests the following version of Problem 3.

Problem 3b. Let d(w, p), 0 < p < 1. Is there a positive and contractive comple-

mented sublattice of d(w, p) linearly homeomorphic to lp1

6. The representation of d(w, l/k) for 1 < k G N. In this section we give a

representation of the dual of d(w, l/k), 1 < k G N, and also a representation of

the Mackey completion of this space. These representations seem to be useful in

the study of the d(w,p)'s structure. Remark first that the dual of d(w,p), 0 <p <

1, is

Íd(w,p)' = | (o,),°l,: 2 |ai| \K(o\ < 0° for every permutation 7T

I i = i

of the integers and for every a + (a,)~=, G d(w, p) \. (6.1)

Then we have

Proposition 6.1. Let d(w,p), wherep = l/k, k > 1. Then

d(w,P)' n c0 = c0 n j (6,)r_,: sup I 2 ¿,*)/ ( 2 *,] < oo J = E,

IHi;,„ = sup(2¿,*)/(2 *,)> (6.2)

w/iere 6 = (6,-)," j G c0 and b* = (p,*),0^ is a decreasingly rearrangement of b.

Proof. Let b G d(w,p)' n c0, i„ = 2"_, o* and j„ = 2"_, wt. Suppose first that

(6.3)n S

Since lim„ sn = + oo, it follows that lim„ tn = + oo.

Let «0 = 0 and sn = 0. If m G N is fixed, then (6.3) implies that

t„ - tmn msup —-~- = + oo. (6-4)

Choosing nx arbitrarily we can find n2 G N, n2 > nx such that

s„ — s„ > s„ — s„ (since lim s„ = oo),"2 "i "i "o V n '

t„2-,n¡>22k-x(sn2-sn)k (by (6.4)).

By induction we get an increasing sequence of integers («,-)," x such that

1+1 1 1 I-i (6.->)for> =1,2,_

^+, - 'n, > (J + 1)2*"'(V, ~ S)" (6.6)



Now let «,_, < i < rtj and y, = j 2k(sn — s ) k. Then y = (y¡)fLx is a decreasing

sequence (by (6.5)) and moreover•» 1-

2 ^y* • w, = 2 21 = 1 7=1 f-l$_| + l

= 2 ¡/J2 < oo,

that is_y G d(w,p). On the other hand

2 ytb? =2 2 i-ii=i = 1 , = «,_, + ! y (j - j )

i (\ - V.) 7

> (by (6.6)) > 2 2 -=+oo,1r-|7

7=1 i = ny_, + l 7

which contradicts the fact that b G í7(w, 77)'. Consequently (6.3) is not true and

then

Xk = sup — < + 00. (6.7)" sk

Hence d(w,p)' n c0 G E.

Now let o G E. If the decreasing sequence^ = (y¡)fL\ e d(w,p), then

n-l

2 M* - 2 t,(y, - yi+i) + v«1=1 1-1

< (by (6.7)) < Xk 2 sk(y, -yi+x) + sfrn1 = 1

<(by(5.1))<A,(2>',,/A:->v1)

Hence \\b\\'pw = ||o*||^)K, < sup„ tn/sk and b G d(w,p)' n c0. On the other hand,

for every e > 0, there is n G N with t„/sk > Xk — e. Let

.V,if 1 < n,

0 if i > n.

Then IMI^ = 1 and \\b\\'p,w = ||6%,„ > 2?_,M = /„/*„* > Xk - e. e being ar-

bitrarily small, it follows that ||o||p)M, = sup„ tn/sk, consequently the equalities (6.2)

are satisfied. □

Corollary 6.2. Letp = l/k, k > I.

(a) Ifd(w,p) +./, then

d(w,p)'= (¿>,) £c0:sup(2¿,*)/(2 »,)

= {(¿>,)r_i: suP(2 v)/(i *.■) <c

< 00

(b) If d(w,p) c /„ then d(w,p)' = lx.

Proof. The case (b) is clear by Proposition 3.4.

(a) Let b G d(w,p)'. If o G c0, then there are e > 0 and («,•)_,", such that \b \ > e

for every j G N. Reasoning as in the proof of (3.3) we get a contradiction. Hence


454 NICOLAE POPA

d(w, p)' c c0, consequently, by (6.2), it follows the first equality. By the last part of

the proof of Proposition 6.1 it follows the second equality. □

We state now the main result of this section:

Theorem 6.3. Let p = l/k < 1 and d(w,p) c£ /,. If there is a positive decreasing

sequence (vX= i e co sucn mat

n i n \ *

2 v,■ ~ 2 w¡) M every n G N (6.8),=i \,=i /

(i.e. there are constants A, B > 0 such that ^(2"_i o,) < (2"_, w,f < 5(2"_, «,.)

for every n G N), then

d(w7ïj^d(v, 1). (6.9)

Proof. Theorem 11 of [4] says that d(v, 1)' = {(è,),",: sup„(27_! of)/(2"_, o,)

< oo }. Then by the hypothesis and by Corollary 6.2(a) it follows that

d(w,p)' = d(v,l)'. (6.10)

It is easy to see that the canonical basis (xX-i or d(w' P) is a symmetric basis of

d(w,p) too. Then, denoting by d(w,p)x the Köthe dual of d(w, p), that is the space

{(0,)°!,: 2°°_,|a,o,| < +00 for every a = (ai)°°=x £\d(w,p)), it follows that d(w,p)x

= d(w,p)x = d(w,p)' = </(w,//)'. Consequently i/(w,/7)xx = d(w,p)'x = (by (6.10))

= d(v, l)'x = (by Corollary of Theorem 10 of [4]) = d(v, l)xx. We shall show that

d(v, l)xx = d(v, 1).

A Köthe result (see §30, p. 5 of [6]) says that for a Banach sequence space E,

£xx _ g ¿f an(j onjy jj £■ js Yêaîy sequentially complete. Hence, it suffices to

show that d(w, 1) is weakly sequentially complete. By Theorem l.c.10 of [8] it

follows that we must show that c0 <£ d(v, 1). If c0 c ¿(u, 1), Corollary 17 of [3]

implies that c0 contains a complemented subspace linearly homeomorphic to /,,

which is a contradiction. Hence d(w,p)xx = d(v, l)xx = d(v, 1), consequently

d(w,p) c d(v, 1) c i/(w,/7)xx. By Proposition 3.3 it follows that every order inter-

val [0, x] c d(w,p) is compact. On the other hand Theorem II. 5.10 of [11] shows

us that in a Banach lattice E the following assertions are equivalent:

(1) The norm of E is order continuous (i.e. if (ya)aÂ is a downward directed set

in E with infa ya = 0, then lima||>'c<|| = 0).

(2) Every order interval [0, x] c E is a(E, £")-compact.

(3) E is an order ideal in E".

Finally, a theorem of Ando (see Proposition IV. 11.1 of [11]) says that a Banach

lattice E has an order continuous norm if and only if every closed order ideal of E

is the range of a positive projection from E.

Consequently d(w, p) is a complemented subspace, with a symmetric basis (y„)„,

of the space d(w,p)" = d(w,p)xx = d(v, 1). Since d(w,p) sé /„ by the Corollary 14

of [3] it follows that d(w, p) tu d(v, 1). □

Remark 6.4. (1) d(w, l/k) with k > 1 and wn = l/n", where (k - l)/k < a <

1, are examples of the spaces d(w, l/k) which verify the conditions of Theorem

6.3. Indeed, if a = 1, we take in Proposition 3.7, y = 0, b = 1 and n, = j for every

j G N. Hence d(w, l/k) +. /,.



If (k - l)/k < a < 1, let 0 < y < l/(k - 1), rtj = jx/a for every j G N, and

b = supj[(j + l)x/a -j]/a]/jy < + oo. Then, by Proposition 3.7, it follows that

d(w, l/k) +_ /,. Suppose now again that a = 1 and put

1 if i <[ek~x] + 1,

if; >[**-»]+ i(log 0 '

where [e* '] is the entire part of ek '.Then

" f"(iogx)k~l , n .k ( ¿ \* .¿j u, ~ I J- dx ~ (log n) — ¿i wi\ ror every n.1=1 J3 x \,= i /

Moreover limn v„ = 0 and vn > vn+x for every n G N.

If (k - l)/k < a < 1, then we put vn = l/nß, where ß = k(a - (k - l)/k) <

1 for every n G N. Clearly limn u„ = 0 and (vX-1 is a decreasing sequence.

Moreover

" r" dx I " \k.2 °<~/ ^~«1_/3~«M1_a)~ 2 w, for every/j G N.i=l ix \ ; =i /

(2) We do not know if the condition (6.8) is superfluous.

Let us mention the following problem:

Problem 4. Let p = q/k, 1 < q < k. Is there a positive decreasing sequence

(©,)£, G c0 \ /, such that d(w, p) sa d(v, q)1

More generally:

Problem 4a. Let 0 <p < l,p ¥= l/k for any /c G N. Is d(w,p) reflexive?

Remark that a positive answer to Problem 4a implies a positive answer to

Problem 3 forp =£■ l/k.

References

1. Z. Altshuler, P. G. Cassaza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces,

Israel J. Math. 15 (1973), 140-155.

2. G. Bennett, An extension of the Riesz-Thorin theorem, Lecture Notes in Math., vol. 604,

Springer-Verlag, Berlin and New York, 1977.

3. P. G. Cassaza and B. L. Lin, On symmetric basic sequences in Lorentz sequence spaces. II, Israel J.

Math. 17(1974), 191-218.

4. D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. 16 (1966), 85-106.

5. N. J. Kalton, Orlicz sequence spaces without local convexity, Proc. Cambridge Philos. Soc. 81

(1977), 253-277.6. G. Köthe, Topological vector spaces. I, Springer-Verlag, Berlin and New York, 1969.

7. J. Lindenstrauss and A. Peltzyñski, Absolutely summing operators in £,p-spaces and their applica-

tions, Studia Math. 29 (1968), 275-326.

8. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I. Sequence spaces, Springer-Verlag,

Berlin and New York, 1977.

9. B. Maurey, Théorèmes de factorisation pour ¡es operateurs linéaires à valeurs dans les espaces Lp,

Astérisque 11 (1974), 1-163.

10. S. Rolewicz, Metric linear spaces, PWN, Warszawa, 1972.


456 NICOLAE POPA

11. H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York,

1974.

12. W. J. Stiles On properties of subspaces of lp, 0 <p < 1, Trans. Amer. Math. Soc. 149 (1970),

405-415.

13. B. Walsh, On characterizing Kothe sequence spaces as vector lattices, Math. Ann. 175 (1968),

253-256.

Department of Mathematics, INCREST, Bucharest 77538, Romania

Current address: Mathematisches Institut, Universität des Saarlandes, Saarbrücken 6600, West

Germany

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