On Stability Conditions of Operator Semigroups · semigroups we refer the reader to Engel Nagel's monograph [20] and [2]. One of the most studied parts of the theory of linear operator

On Stability Conditions of Operator Semigroups

Zoltán Léka

Bolyai Institute

University of Szeged

A thesis submitted for the degree of

Doctor of Philosophy

2009

mailto:[email protected]

http://www.math.u-szeged.hu

http://www.u-szeged.hu

Acknowledgements

First, I would like to express my deep gratitude to Prof. László Kérchy for

his valuable suggestions during the preparation of this thesis, which helped

make the proofs clear and concise, and my colleagues at the Institute for

their useful comments.

Next, I would like to thank Prof. Jaroslav Zemánek for providing me with an

inspiring atmosphere, and for the warm hospitality shown when I spent two

months at the Banach Center, in Warsaw. Then I must express my thanks

to Prof. László Zsidó for several interesting discussions on dierent topics

in mathematics, to Prof. C.J.K. Batty who sent me doctoral dissertations

related to my work, and to David P. Curley for reading this manuscript and

improving its style.

Last, but not least, I must mention my parents, and Eszter who have stood

by me all these years. This thesis is dedicated to them.

Abstract

The principal aim of this thesis is to prove stability results for semigroups

of operators. This study is based on two central operator theoretical re-

sults, namely the ArendtBattyLyubichVu theorem and the Katznelson

Tzafriri theorem. Both theorems help provide a connection between the

spectral properties of semigroups of operators and their asymptotic be-

haviour.

We shall give an extension of the ABLV theorem for certain unbounded

representations that have a regular norm-function. Then we will charac-

terise those C0-semigroups whose norm-function is topologically regular. In

addition, we shall prove a KatznelsonTzafriri type theorem for discrete

one-parameter semigroups in Hilbert spaces.

Contents

1 Introduction 1

1.1 Asymptotic behaviour of operator semigroups . . . . . . . . . . . . . . . 1

1.1.1 The ABLV theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Representations and regularity . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Topological regularity . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The KatznelsonTzafriri theorem . . . . . . . . . . . . . . . . . . . . . . 4

2 Representations with regular norm-functions 6

2.1 Almost convergence on semigroups . . . . . . . . . . . . . . . . . . . . . 6

2.2 Existence of the limit functional . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Spectra of representations . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Regularity and isometric representations . . . . . . . . . . . . . . . . . . 22

2.5 The stability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 C0-semigroups and regularity 25

3.1 Topologically invariant means . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 An immediate application . . . . . . . . . . . . . . . . . . . . . . 30

3.2 C0-semigroups and topological regularity . . . . . . . . . . . . . . . . . . 33

3.3 The regularity constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 One consequence of regularity . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 The converse statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.6 An example for a gauge function . . . . . . . . . . . . . . . . . . . . . . 45

4 A KatznelsonTzafriri type result 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Preliminaries and the main result . . . . . . . . . . . . . . . . . . . . . . 48

iv

CONTENTS

5 Summary 54

5.1 Representations of locally compact commutative semigroups . . . . . . . 54

5.2 C0-semigroups and topological regularity . . . . . . . . . . . . . . . . . . 56

5.3 A KatznelsonTzafriri type theorem in Hilbert spaces . . . . . . . . . . . 56

6 Összefoglalás 58

6.1 Lokálisan kompakt, kommutatív félcsoportok reprezentációi . . . . . . . 58

6.2 C0-félcsoportok és topologikus regularitás . . . . . . . . . . . . . . . . . 60

6.3 KatznelsonTzafriri típusú tétel Hilbert tereken . . . . . . . . . . . . . . 61

References 63

v

Chapter 1

Introduction

1.1 Asymptotic behaviour of operator semigroups

Consider the well-posed abstract Cauchy problem

(ACP)

u(t) = Au(t) (0 ≤ t),u(0) = x,

where A is a densely dened, closed operator acting on a complex Banach space X and

x ∈ X. We say that the family of bounded linear operators (T (s))s≥0 acting on X is

a C0-semigroup if T (0) = I, T (s + t) = T (s)T (t) for all s, t ≥ 0 and the mapping

s 7→ T (s)x is continuous for every x ∈ X. Then the (classical and mild) solutions of the

equation (ACP) are derived from a C0-semigroup (T (s))s≥0. The operator semigroup

(T (s))s≥0 uniquely determines the operator A, which is called the innitesimal generator

of (T (s))s≥0; that is,

Ax = limh↓0

T (h)x− xh

when the limit exists. The set of vectors for which the above limit does exists is called

the domain of A and it shall be denoted by D(A). For a comprehensive study on C0-

semigroups we refer the reader to EngelNagel's monograph [20] and [2].

One of the most studied parts of the theory of linear operator semigroups is the

asymptotic behaviour of the semigroup. We say that a C0-semigroup (T (s))s≥0 is stable

if T (s) tends to zero, if s→∞ in the strong operator topology. A systematic account of

the stability theory of linear operator semigroups is presented in [54], [2] and [13]. The

uniform and weak stability properties of the semigroup can be also described in terms of

1

1.1 Asymptotic behaviour of operator semigroups

the operator norm topology and the weak operator topology. On recent developments

of weak stability theorems we refer the reader to the articles [18] and [19]. A detailed

overview of these results can also be found in [61].

1.1.1 The ABLV theorem

In 1988, W. Arendt and C.J.K. Batty [1], and Y. Lyubich and Q. P. Vu [50], inde-

pendently and simultaneously, proved a famous stability result on bounded continuous

one-parameter semigroups of operators. Let us consider a C0-semigroup T of bounded

linear operators on a complex Banach space X. We will denote the algebra of bounded

linear operators acting on X by L(X). The generator of T shall be denoted by A, and

σ(A), σp(A∗) will stand for the spectrum of A and the point spectrum of its adjoint A∗,

respectively. We say that T is bounded if supt∈R+‖T (t)‖ < ∞. Next, we will present

the well-known ABLV theorem.

Theorem 1.1.1. Let T be a bounded C0-semigroup on X having the generator A, and

suppose that σ(A) ∩ iR is countable and σp(A∗) is empty. Then lims→∞ ‖T (s)x‖ = 0for every x ∈ X.

In the Hilbert space setting, B. Sz®kefalvi-Nagy and C. Foias [63] showed earlier

that if T is a completely non-unitary contraction and the peripheral spectrum of T has

zero Lebesgue measure, then the operator T is stable (i.e. Tn → 0 strongly). This

result remains valid in the continuous one-parameter case as well, due to a result by

L. Kérchy and J. Neerven [33].

The ABLV theorem has been investigated by many authors and quite a few general-

izations of the theorem have been proved for bounded and unbounded representations

of suitable, locally compact abelian semigroups (see [4], [8], [6], [36], [37]).

The rst abstract version of the theorem was given by Batty and Vu [8]. Let G

be a locally compact group and let S be a σ-compact subsemigroup of G, with non-

empty interior and with S − S = G. Let S∗ and S∗u stand for the space of non-zero

continuous bounded homomorphisms of S into C, and of S into the unit circle of C,respectively. Next, consider S with the restriction of a Haar measure on G. We can

dene for f ∈ L1(S) and χ ∈ S∗ the integral

f(χ) :=∫Sf(s)χ(s) ds.

2

1.2 Representations and regularity

Let us assume that the functions f (f ∈ L1(S)) separate the points of S∗ from each

other and from 0. By a representation T of S, we mean a strongly continuous semigroup

homomorphism T : S → L(X). For any bounded representation T of S and f in L1(S),

we shall dene

f(T ) :=∫Sf(s)T (s) ds.

In their paper [8] Batty and Vu introduced the spectrum of T in the following way:

the spectrum Sp(T ) of T is the set of all characters χ in S∗ such that |f(χ)| ≤ ‖f(T )‖holds for every f ∈ L1(S). The unitary spectrum of T is Spu(T ) := Sp(T ) ∩ S∗u. Theunitary point spectrum Puσ(T ∗) of T ∗ is the set of all χ ∈ S∗u, for which there exists

a non-zero φ ∈ X∗ such that T ∗(s)φ = χ(s)φ (s ∈ S). After these preliminaries, the

following was proved in [8].

Theorem 1.1.2. Let T be a bounded representation of S such that Spu(T ) is countableand Puσ(T ∗) is empty. Then T (s)→ 0 strongly (through S).

We note that rst Vu [66] proved a weighted version of the ABLV theorem. Later,

C.J.K. Batty and S. Yeates [6], [67] gave a detailed study on the spectral theory and

stability of non-quasianalytic representations. A more recent article [3] presents a very

general result in the spirit of [6], but it uses a dierent approach.

1.2 Representations and regularity

Operators with regular norm-sequences were characterized by L. Kérchy and V. Müller

(see [38], [43]). These operators lead to useful generalizations of classical theorems in

operator theory. They were used in [40], [41] to establish a satisfactory symbolic calculus

for generalized Toeplitz operators and to prove results for their elementary and reexive

hyperplanes. In the papers by G. Cassier [11], [12] the concept of regularity was also

used to derive new results on similarity problems. Discrete representations with regular

norm-function were studied in [36], [37], [39] in connection with unbounded versions of

the ArendtBattyLyubichVu stability theorem and the KatznelsonTzafriri theorem.

In this thesis, we shall extend the method, which appears in Kérchy's papers ([36],

[37]) on discrete abelian semigroups, to topological semigroups; that is, we shall prove

stability results for unbounded representations having a regular norm-function. The

stability results we derived are quite similar to [6], but the main dierences lie in the

3

1.3 The KatznelsonTzafriri theorem

way we express the stability of the semigroup and its norm-conditions. Our proof follows

the usual scheme and relies on results concerning isometric representations proved in

[8]. The originality of our approach appears in the usage of almost convergence de-

ning regular norm-behaviour and in the systematic study of representations with this

property. This study is a joint work with L. Kérchy that was published in [42].

In the following chapter we will give a brief summary of the concepts and results

of amenability. The most important tools for us are invariant means which are used

to dene regularity and Følner's condition which characterises amenability (Theorem

2.1.5). After this, we will prove that every representation with regular norm-behaviour

determines a unique character on S. This fact makes it possible for us to introduce the

peripheral spectrum of such a representation. At the end of Chapter 2 we will present

and prove an extension of the ABLV theorem.

1.2.1 Topological regularity

In Chapter 3 we shall study a slightly dierent concept of regularity. We recall that reg-

ularity is dened by means of invariant means, but we were able to use also a (generally)

strictly smaller class of means, the topologically invariant means, to dene another kind

of regularity property of the norm function of representations. We should mention here

that counterparts of the main results in Chapter 2 remain valid with this approach. (In

fact, the main results like the existence of the limit functional and the associated iso-

metric representation can be proved in a similar way.) In discrete semigroups these two

concepts are the same because the classes of these means coincide in the discrete case.

We shall give a detailed exposition of the second alternative on the real half-line (see

[46]). We will also give a characterization of C0-semigroups with topologically regular

norm-function in the spirit of [43].

Our hope is that this result will help provide a better understanding of the regularity

property on the real half line.


In this thesis we will also study another type of stability result. The KatznelsonTzafriri

theorem is a general operator theoretical result which is related to the ABLV theorem.

(The interested reader should consult [23] to learn more about this connection.) Now,

4


let A(T) denote the set of continuous functions on the unit circle T whose Fourier series

is absolutely convergent, and let A+(T) be the set of functions in A(T) whose Fourier

coecients with negative indices vanish. The algebra A(T) is a Banach algebra with

the norm ‖f‖ =∑∞

n=−∞ |f(n)| (where f ∈ A(T) and f(n)∞n=−∞ are the Fourier

coecients of f). We say that an f ∈ A+(T) is of spectral synthesis with respect to a

closed set E ⊆ T if there exists a sequence fnn ⊂ A(T) such that each fn vanishes in

a neighbourhood of E and limn→∞ ‖fn − f‖ = 0. The KatznelsonTzafriri theorem is

the following.

Theorem 1.3.1. Let T be a power-bounded operator on the Banach space X, and let

f ∈ A+(T), which is of spectral synthesis with respect to the peripheral spectrum of T.

Then

limn→∞

‖Tnf(T )‖ = 0.

The important special case of f(z) = z − 1 was studied earlier by Esterle [22]. He

proved that limn→∞ ‖Tn+1−Tn‖ = 0 if and only if σ(T )∩T ⊆ 1. A simple corollary

of this statement is the classical result of Gelfand that an operator T with σ(T ) = 1and supn∈Z ‖Tn‖ < ∞ is the identity. For further details on this statement, see [68].

Esterle's result was later obtained independently and generalized by Y. Katznelson and

L. Tzafriri. Using this result, they gave a new proof and an extension of the zerotwo

law in [34].

In Chapter 4 we shall prove that the assumption made in the KatznelsonTzafriri

theorem can be weakened in Hilbert spaces, and we shall provide a complete charac-

terization of the condition limn→∞ ‖TnQ‖ = 0 whenever Q commutes with T. These

results are based on [47].

We note that many extensions of the KatznelsonTzafriri theorem were proved in

the discrete case as well as in the continuous case using dierent methods. For details,

see [8], [24], [33], [51], [52], [65] and related survey articles [4], [13]. However, we should

also remark that all former extensions of the KatznelsonTzafriri theorem are related

to bounded functional calculi of T or elements of the Banach algebra generated by T.

5

Chapter 2

Representations with regular

norm-functions

In Section 2.1 we shall introduce the concept of almost convergence (dened for se-

quences in [48]) in terms of invariant means, which will be needed to dene the limit

functional in Section 2.3 for unbounded representations satisfying a certain regularity

condition. It turns out that this limit functional is intrinsic to the representation itself

and independent of any choice of gauge function involved in the regularity condition.

This allows us to dene the peripheral spectrum in Section 2.4 along with various spec-

tral notions of representations with a regular norm-function. In the bounded case the

denition coincides with the one introduced by Batty and Vu [8], and Lyubich [49]. In

Section 2.5 we will generalize a well-known method associating an isometric represen-

tation to a representation with a regular norm-function. With these results at hand we

will give, in Section 2.6, a new extension of the ABLV theorem.

2.1 Almost convergence on semigroups

Consider an abelian semigroup (S,+). For any s ∈ S and ω ⊆ S set ω s := s′ ∈ S :

s+s′ ∈ ω. The translation of a function f : S → C by s ∈ S is the mapping fs : S → Cdened by fs(s′) := f(s + s′) (s′ ∈ S). Let (S,Ω, λ) be a σ-nite measure space on S

such that (i) ω s ∈ Ω whenever ω ∈ Ω and s ∈ S, and (ii) λ(ω) = 0 (ω ∈ Ω) implies

λ(ω s) = 0 for all s ∈ S. Obviously, S = Rn+ with the Lebesgue measure and S = Zn+

with the counting measure satisfy these conditions. Let L∞(λ) denote the Banach space

6


of all essentially bounded, complex-valued, measurable functions with the usual norm.

Lemma 2.1.1. Under the conditions (i) and (ii), τs : L∞(λ) → L∞(λ), f 7→ fs is a

well-dened linear mapping with ‖τs‖ = ‖τs1‖ = 1, for every s ∈ S.

Proof. Let χω be the characteristic function of ω ∈ Ω. Since χω is measurable we can

infer from (i) that (χω)s = χωs is also measurable. Using the usual approximation

method we get that fs is measurable for any f ∈ L∞(λ). To see that the translation is

well dened on L∞(λ) suppose that f1 = f2 a.e.; then by (ii) we have (f1)s = (f2)s a.e..Obviously, ‖fs‖∞ ≤ ‖f‖∞. Since ‖1‖∞ = 1 and τs1 = 1, we obtain ‖τs‖ = ‖τs1‖ =1.

The set M(S, λ) := m ∈ L∞(λ)∗ : ‖m‖ = m(1) = 1 is called the set of means on

S with respect to λ. The HahnBanach theorem implies that M(S, λ) 6= ∅. A functional

m ∈ M(S, λ) is called an invariant mean if m(fs) = m(f) holds for every f ∈ L∞(λ)

and s ∈ S; the set of all invariant means with respect to λ shall be denoted by M(S, λ).

It is well known that there are invariant means on abelian groups, and this can be

readily extended to semigroups. For the sake of clarity we shall now sketch the proof.

Proposition 2.1.2. M(S, λ) is not empty.

Proof. Since M(S, λ) is a convex, weak-∗ closed subset of the unit ball of L∞(λ)∗, itfollows that M(S, λ) is a weak-∗ compact set. Taking an m ∈ M(S, λ), for any s ∈ S we

have m τs ∈ M(S, λ), and T (s) : M(S, λ)→ M(S, λ),m 7→ m τs is an ane, weak-∗continuous mapping. The MarkovKakutani theorem (see e.g. [14, theorem V.10.1])

implies the existence of a common xed point m0 ∈ M(S, λ). Then it immediately

follows that m0 is an invariant mean.

Now consider a locally compact, Hausdor abelian group G. Let S be a closed

subsemigroup of G with non-empty interior S such that S−S = G and S∩(−S) = 0.By denition, for any s1, s2 ∈ S, s1 s2 if s2 − s1 ∈ S. In this way we obtain an

inductive partial ordering on S, hence S is a directed set. We say that a function

f : S → C tends to 0 at innity: lims f(s) = 0, if for every ε > 0 there exists an s0 ∈ Ssuch that |f(s)| < ε whenever s0 s. Let µ denote the restriction of the Haar measure

µ on G to S. It is clear that the conditions (i) and (ii) of Lemma 1 are satised for µ;

indeed, ω s = (ω − s) ∩ S. The (invariant) means with respect to µ are simply called

(invariant) means on S; that is L∞(S) := L∞(µ), M(S) := M(S, µ),M(S) := M(S, µ).

7


Throughout this thesis a function f dened on S will also be treated as a function on

G that is zero outside S.

Denition 2.1.3. A net Kλλ∈Λ of compact subsets of G with nonempty interior is

a strong Følner net for G if

(i) Kλ1 ⊆ Kλ2 whenever λ1 ≤ λ2,

(ii) G =⋃Kλ,

(iii) µ((x+Kλ) 4 Kλ)/µ(Kλ) → 0 (as λ → ∞) uniformly when x runs through

compact sets. (Here and in the sequel 4 stands for the symmetric dierence.)

A net Kλλ∈Λ of compact subsets of G with nonempty interior is called a Følner net

for G if (iii) is satised.

Example 2.1.4. We can give some simple examples here.

• if G = Z, the sets 0, ..., n (n ∈ N) (or −n, ..., n) form a Følner sequence for Z;

• the intervals [0, n] (n ∈ N) (or [−n, n]) provides a Følner sequence for R.

We recall that an arbitrary locally compact group G is called amenable if there exists

an invariant mean on G. The following theorem is a useful and deep characterization of

amenable groups (for details of its proof, see [56, Theorem 4.16]).

Theorem 2.1.5. A locally compact group G is amenable if and only if there exists a

strong Følner net for G. If G is σ-compact, then G is amenable if and only if there

exists a strong Følner sequence for G.

We shall assume in the sequel that G is σ-compact. Since G is abelian by our

assumption, the MarkovKakutani theorem implies that G is amenable (see the proof

of Proposition 2.1.2), thus using the characterization theorem we obtain the existence of

a strong Følner sequence Knn∈N on G.We shall translate this sequence to the interior

of S, preserving property (iii), by means of the following two lemmas.

Lemma 2.1.6. If Knn is a Følner sequence for G then Kn + snn is also a Følner

sequence for G, for every sequence sn∞n=1 in G.

Proof. Using the translation invariance of the Haar measure, for any pair of measurable

sets B1, B2 and g ∈ G, we have µ((B1+g) 4 (B2+g)) = µ(B1 4 B2); hence Kn+snnis also a Følner sequence.

8


Lemma 2.1.7. Let K be a compact set in G. Then there exists an s ∈ S such that

K + s ⊆ S.

Proof. Since by the assumption G = S − S, for any g ∈ G and s0 ∈ S we have

g − s0 = s′ − s′′ with some s′, s′′ ∈ S. Since s0 + s′ ∈ S clearly holds, we nd that

G = S − S is satised. The family S − s : s ∈ S forms an open covering of K

and so there exist s1, . . . , sn ∈ S such that K ⊆⋃ni=1(S − si). Setting s =

∑ni=1 si the

relation si s implies S− si ⊆ S− s for every 1 ≤ i ≤ n. It follows that K ⊆ S− s,that is K + s ⊆ S.

An immediate consequence of Lemma 2.1.6 and Lemma 2.1.7 is that there must be

a Følner sequence in the interior of S.

We shall need yet another result on topological semigroups that will be used in this

chapter. Recall that a function f on S is called locally bounded if it is bounded on the

compact subsets of S. Let Cc(S) denote the set of continuous functions with compact

support in S.

Lemma 2.1.8. Let K ⊆ S be a compact set of G. If f is a measurable, locally bounded

function on S and g ∈ L∞(S) then the convolution

(f ∗ g)(s) :=∫Kf(s+ t)g(t) dµ(t)

is continuous on S.

Proof. Assuming f ∈ Cc(S), we can easily verify that f is uniformly continuous on

S; that is, for every ε > 0, there exists an open set U in G containing 0 such that

|f(s′) − f(s)| ≤ ε whenever s′ − s ∈ U and s′, s ∈ S. Thus we may infer that f ∗ g is

continuous.

Suppose now that f is any locally bounded, measurable function. We can choose

a compact neighbourhood K1 of 0 in G such that K + K1 ⊆ S. Indeed, every s ∈ Kis an inner point of S hence there exists an open neighbourhood Vs of 0 (in G) such

that s + Vs ⊆ S. Furthermore, choosing open neighbourhoods Ws of 0 such that

Ws +Ws ⊆ Vs, the family s+Ws : s ∈ K forms an open covering of K and therefore

has a nite subcovering s1 +Ws1 , ..., sn+Wsn . Let us form the open setW :=⋂ni=1Wsi ;

then K + W ⊆ (⋃ni=1 (si +Wsi)) + W =

⋃ni=1(si + Wsi + W ) ⊆

⋃ni=1(si + Vsi) ⊆ S.

Since G is a locally compact Hausdor space there is a compact neighbourhoodK1 ⊆Wof 0, hence K2 := K +K1 ⊆ S. It is clear that K2 is also compact.

Let us take an arbitrary ε > 0, and x a y ∈ S. Since f is in L1(S, χK2+ydµ), thereexists an h ∈ Cc(S) such that

∫K2+y |f(t)− h(t)| dµ(t) < ε. (Notice that the restricted

9


Haar measure µ is regular on the σ-compact S; see [57, p. 238].) We already know

that h ∗ g is continuous. Hence we can choose a neighbourhood U of 0 in G such that

U ⊆ K1 and |(h ∗ g)(y)− (h ∗ g)(v)| < ε is true whenever v ∈ (y + U) ∩ S. Then

|(f ∗ g)(y)− (f ∗ g)(v)| ≤∫K|fy − hy||g| dµ+ |(h ∗ g)(y)− (h ∗ g)(v)|

+∫K|hv − fv||g| dµ

<

∫K+y|f − h||g| dµ+ ε+

∫K+v|h− f ||g| dµ

≤ 2‖g‖∞∫y+K2

|f − h| dµ+ ε

< 2ε‖g‖∞ + ε,

and the lemma follows.

Our intention is to prove stability results and discuss the asymptotic properties of

operator semigroups. In most cases we shall use weaker notions of the usual convergence

of orbits, which requires invariant means. First we will proceed with a study of almost

convergent functions.

Denition 2.1.9. A function f ∈ L∞(S) is called almost convergent if the set m(f) :m ∈ M(S) is a singleton. We shall use the notation a-lim f = c whenever m(f) = c

for all m ∈M(S).

A useful property of almost convergence is described in the following proposition.

Proposition 2.1.10. If f ∈ L∞(S) is almost convergent with a-lim f = c and Knnis a Følner sequence on S then

limn→∞

1µ(Kn)

∫Kn

fy dµ = c

uniformly with respect to y ∈ S.

Proof. Let us assume that there exist an ε > 0, a strictly increasing sequence nkk of

positive integers, and a sequence ykk ∈ S such that∣∣∣∣∣ 1µ(Knk)

∫Knk

fyk dµ − c

∣∣∣∣∣ > ε

is true for every k ∈ N.

10


Applying Lemma 2.1.6 to the subsequence Knkk we nd that Ck := Knk +ynkkis also a Følner sequence. For every k ∈ N, let us consider the linear functional ϕk on

L∞(S) dened by

ϕk(g) :=1

µ(Ck)

∫Ck

g dµ , g ∈ L∞(S).

Obviously, ‖ϕk‖ = ϕk(1) = 1 is valid. By the BanachAlaoglu theorem there exists a

weak-∗ cluster point m0 of the sequence ϕkk. Clearly, ‖m0‖ = m0(1) = 1. Further-more, for every g ∈ L∞(S) and y ∈ S, we have

|ϕk(g)− ϕk(gy)| =1

µ(Ck)

∣∣∣∣∫Ck

g dµ−∫Ck+y

g dµ

∣∣∣∣≤ µ((Ck+ y)4 Ck)

µ(Ck)‖g‖∞ → 0, as k →∞,

whence m0(g) = m0(gy) follows; that is, m0 is an invariant mean.

Since |m0(f)−c| ≥ ε by our assumption we conclude that f cannot almost converge

to c.

Remark 2.1.11. This result suggests that almost convergence has some natural con-

nection with abstract ergodic theory. The interested reader should see H.A. Dye's paper

[17] on abstract weak mixing and almost convergence.

We shall also introduce a stronger form of almost convergence.

Denition 2.1.12. We say that a function f ∈ L∞(S) almost converges in the strong

sense to a complex number c, if a-lim |f−c| = 0. In that case we shall use the notation:

as-lim f = c.

In the lemma below we shall prove a multiplicative property of as-lim. This state-

ment is analogous to [36, Lemma 1].

Lemma 2.1.13. For every f ∈ L∞(S), the following statements are equivalent:

(i) as-lim f = c,

(ii) m(fg) = cm(g) for every g ∈ L∞(S) and m ∈M(S).

Proof. (i) ⇒ (ii): Every invariant mean m ∈ M(S) is a positive functional on L∞(S).Given any h ∈ L∞(S), let α be a complex number with the properties |α| = 1 and

|m(h)| = αm(h). Since |h| − Re (αh) ≥ 0, it follows that m(|h|) ≥ m(Re(αh)) =Re m(αh) = |m(h)|. Therefore

|m(fg)− cm(g)| = |m(fg − cg)| ≤ m(|fg − cg|) ≤ ‖g‖∞m(|f − c1|) = 0.

11

2.2 Existence of the limit functional

(ii) ⇒ (i): Let g(s) := |f(s) − c|(f(s) − c)−1 if f(s) 6= c and 0 otherwise. Then

g ∈ L∞(S), and m(|f − c1|) = m((f − c)g) = m(fg)− cm(g) = 0.

It is easy to check that convergence implies almost convergence in the strong sense.

Let us consider the function f ∈ L∞(R+), where f(s) := 1 if n ≤ s < n + 1 and n

is even, while f(s) := −1 otherwise. Since f + f1 is identically zero, we infer that

a-lim f = 0, but lims→∞ f(s) does not exists. It is also clear that f does not converge

in the strong sense. Let us now take the function g ∈ L∞(R+), where g(s) := 1 if

2n ≤ s < 2n + 1 (n ∈ N), while g(s) := 0 otherwise. For any a ∈ R+ and N ∈ N,let Ga,N := N−1

∑Nk=1 ga+k. Since m(Ga,N ) = m(g) holds for every invariant mean

m ∈ M(R+), and since ‖Ga,N‖∞ can be arbitrarily small, we conclude that as-lim

g = 0. On the other hand, lims→∞ g(s) does not exist.


The regular norm-behaviour of representations will be dened in terms of gauge func-

tions.

Denition 2.2.1. We say that p : S → (0,∞) is a gauge function if it is measurable

and, for every s ∈ S, ps/p ∈ L∞(S) almost converges in strong sense to a positive

number cp(s). The function cp is called the limit functional of the gauge function p.

If p : S → (0,∞) is a gauge function then, by Lemma 2.1.13 the equation

ps1+s2

p=

(ps1)s2ps2

· ps2p

implies that cp(s1 + s2) = cp(s1)cp(s2) (s1, s2 ∈ S); that is, the limit functional cp of p

is multiplicative.

Next we shall prove a basic property of the limit functional which seems to be

important for deriving results later on.

Lemma 2.2.2. Let p be a gauge function with p(s) ≥ 1 for s ∈ S. Then cp(s) ≥ 1 for

every s ∈ S.

Proof. We can see from Proposition 2.1.10 that, for every s ∈ S,

1µ(Kn)

∫Kn

(ps/p)y dµ → cp(s) (n→∞),

12


uniformly with respect to y, where Knn is a Følner sequence in S. Assume that

cp(s0) < 1 for some s0 ∈ S, and let us choose cp(s0) < δ < 1. There exists an n0 ∈ Nsuch that µ(Kn0)−1

∫Kn0

(ps0/p)y dµ ≤ δ is true for every y ∈ S. Now the Jensen

inequality implies that

1µ(Kn0)

∫Kn0

(log ps0+y − log py) dµ ≤ log

(1

µ(Kn0)

∫Kn0

(ps0/p)y dµ

)≤ log δ < 0 (y ∈ S).

Summing the inequalities obtained with the choices y = 0, s0, 2s0, ..., (m− 1)s0 we nd

that1

µ(Kn0)

∫Kn0

log pms0 dµ ≤1

µ(Kn0)

∫Kn0

log p dµ +m log δ

is true for every m ∈ N. This is a contradiction because the left-hand side of the

inequality must be nonnegative, since p ≥ 1, but the right-hand side is negative for

large m.

We recall that the non-zero, continuous homomorphisms of S into the multiplicative

semigroup of C are called the characters of S.

Corollary 2.2.3. Let χ be a character of S such that cp ≤ |χ| ≤ p. Then |χ| = cp.

Proof. It is clear that p = p/|χ| is a gauge function with cp = cp/|χ|. Since p ≥ 1,Lemma 2.2.2 tells us that cp ≥ 1, and so cp ≥ |χ|.

Throughout this thesis X will stands for a complex Banach space and L(X) will be

the algebra of all bounded linear operators on X. A representation ρ of S on the Banach

space X is a mapping ρ : S → L(X) such that

• ρ(0) = I,

• ρ(s+ t) = ρ(s)ρ(t) (s, t ∈ S),

• s 7→ ρ(s)x is continuous for every x ∈ X.

Proposition 2.2.4. Let ρ : S → L(X) be a bounded representation. Then, for every

x ∈ X, ‖ρ(s)x‖ almost converges to zero if and only if lims ‖ρ(s)x‖ = 0.

Proof. Assuming a-lims ‖ρ(s)x‖ = 0 with x ∈ X, by Proposition 2.1.10 we get that, for

any ε > 0, ‖ρ(s0)x‖ ≤ εM−1 holds for some s0 ∈ S, where M := sup‖ρ(s)‖ : s ∈ S.Then ‖ρ(s+ s0)x‖ ≤MεM−1 = ε is true for every s ∈ S, and so lims ‖ρ(s)x‖ = 0. Theconverse implication is trivial.

13


Now we will introduce the concept of regularity.

Denition 2.2.5. The representation ρ : S → L(X) is of regular norm behaviour with

respect to the gauge function p or has p-regular norm-function if ‖ρ(s)‖ ≤ p(s) holds forevery s ∈ S, and a-lims‖ρ(s)‖/p(s) = 0 is not true.

There is a connection between the spectral radius function r(ρ(s)) of a representation

ρ with p-regular norm function and the limit functional cp(s). Namely, the following

inequality always holds (see [37, Proposition 8]):

Lemma 2.2.6. The inequality cp(s) ≤ r(ρ(s)) is true for every s ∈ S.

Proof. Let m ∈ M(S) be an invariant mean such that m(‖ρ(s)/p(s)) 6= 0. Since the

inequalityp(s+ t)p(s)

‖ρ(s+ t)‖p(s+ t)

≤ ‖ρ(t)‖‖ρ(s)‖p(s)

(s, t ∈ S)

holds, we may infer by the positivity of m and Lemma 2.1.13 that

m

(p(s+ t)p(s)

)m

(‖ρ(s+ t)‖p(s+ t)

)≤ ‖ρ(t)‖m

(‖ρ(s)‖p(s)

).

Hence

cp(t) ≤ ‖ρ(t)‖

is satised for every t ∈ S. Since cnp (t) = cp(nt) ≤ ‖ρ(nt)‖ ≤ ‖ρ(t)n‖ if n ∈ N, applyingGelfand's spectral radius formula, we nd that cp(t) ≤ r(ρ(t)).

As we are mostly interested in unbounded representations of S which are bounded

on compact sets, the following assumption seems natural:

Assumption 2.2.7. We shall assume throughout this thesis that for any gauge function

p we have p ≥ 1 and that p is locally bounded on S.

Now we will show that the limit functional cp(s) is a character of the semigroup S.

Theorem 2.2.8. Let p be a gauge function on S which satises Assumption 2.2.7 and

let us assume that there exists a representation ρ : S → L(X) with a p-regular norm-

function. Then the limit functional cp of p is a positive character of S.

Proof. We already know that cp(s′+ s′′) = cp(s′)cp(s′′), s′, s′′ ∈ S, so cp is a homomor-

phism. It remains to show that cp is continuous. First, applying Proposition 2.1.10 and

Lemma 2.1.8, we see that cp is a measurable function because it is a sequential limit of

continuous functions.

14


The Principle of Uniform Boundedness tells us that the function ‖ρ(s)‖ is boundedon compact sets, thus Lemma 2.2.6 implies that the funcion cp(s) is also locally boundedon S. Let us choose a compact setK ⊆ S such that µ(K) > 0. Then α :=

∫K cp(t)dµ(t) ∈

(0,∞). Given any y ∈ S, we have∫Kcp(y + t) dµ(t) =

∫Kcp(y) cp(t) dµ(t) = cp(y)

∫Kcp(t) dµ(t),

whence cp(y) = α−1∫K cp(y + t) dµ(t) follows. Then we can apply Lemma 2.1.8 to

deduce the continuity of the limit functional cp.

We have now arrived at the main result of the section.

Theorem 2.2.9. Let G be a locally compact, σ-compact, Hausdor abelian group with

a closed subsemigroup S such that S − S = G, S ∩ (−S) = 0 and S 6= ∅. If therepresentation ρ : S → L(X) is of regular norm-behaviour with respect to the gauge

functions p and q which satisfy Assumption 2.2.7, then

cp = cq.

Proof. We can see from Lemma 2.2.6 that

cp(s) ≤ r(ρ(s)) ≤ ‖ρ(s)‖ ≤ q(s) (s ∈ S),

and c−1p q is a gauge function by Theorem 2.2.8. Thus by Lemma 2.2.2 we have

1 ≤ cc−1p q = c−1

p cq,

so cp ≤ cq. In a similar way we nd that cp ≥ cq, thus cp = cq.

The above theorem leads to the following denition.

Denition 2.2.10. The function cρ := cp is called the limit functional of the represen-

tation ρ with p-regular norm-function.

It was already shown in [36] that for S = Z+ the limit functional cρ(n) is equal to

r(ρ(n)) (n ∈ Z+). The analogous statement concerning C0-semigroups is valid. To prove

this, we need to state a simple lemma. We say that a representation T : R+ → L(X) is

quasinilpotent if r(T (s)) = 0 holds for every s > 0. (Notice that in the quasinilpotent

case r(T (s)) is not continuous, since r(T (0)) = 1, and so it is not a character.)

Lemma 2.2.11. If the representation T : R+ → L(X) is not quasinilpotent then r(T (s))is a character of R+.

15


Proof. As this statement is well known, we will only sketch a proof. The submulti-

plicativity r(T (s + t)) ≤ r(T (s))r(T (t)) (s, t ∈ R+) and r(T (ns)) = r(T (s))n (n ∈N, s ∈ R+) yield that r(T (s)) > 0 for every s ∈ R+. Furthermore, it can be easily

checked that ω0 := lims→∞ s−1 log ‖T (s)‖ ∈ R exists; see e.g. [20, p. 251]. Thus

r(T (s)) = limn→∞ ‖T (ns)‖1n = exp(s limn→∞(ns)−1 log ‖T (ns)‖) = exp(ω0s) is true

for every s ∈ R+.

Proposition 2.2.12. If the representation T : R+ → L(X) has regular norm-behaviour

(with respect to a gauge function p which satises Assumption 2.2.7), then cT (s) =r(T (s)) (s ∈ R+) holds.

Proof. We know from Lemma 2.2.6 that r(T (s)) ≥ cT (s) > 0 (s ∈ R+), and so T : R+ →L(X) is not quasinilpotent. Hence Lemma 2.2.11 implies that r(T (s)) is a character of

R+. Applying Corollary 2.2.3 we nd that r(T (s)) = cT (s) (s ∈ R+).

However, the following example shows that the spectral radius function and the

limit functional can be dierent.

Example 2.2.13. Let us choose the semigroup R2+ and the weight function w(x, y) :=

ex(1−y) + 1. Next, consider the Banach space Cw of continuous functions f : R2+ → C

which satises the conditions f |∂R2+ ≡ 0 and ‖f‖w := supR2

+|f(x, y)|w(x, y) <∞. The

strongly continuous representation T : R2+ 7→ L(Cw) is dened by

(T (s, t)f)(x, y) :=

f(x− s, y − t) if (x− s, y − t) ∈ R2+,

0 otherwise.

Since lim(x,y)→∞w(x, y) = 1 it follows that, for every (s, t) ∈ R2+,

lim(x,y)→∞

w(s,t)(x, y)/w(x, y) = 1,

hence w is a gauge function (which satises Assumption 2.2.7) with cw ≡ 1. Since w is

submultiplicative, we have ‖T (s, t)‖ ≤ w(s, t) ((s, t) ∈ R2+). Taking a sequence fnn

in Cw satisfying the condition 0 ≤ fn(x, y) ≤ fn(1/n, 1/n) = 1, (x, y) ∈ R2+, where the

support of fn is included in the disc centered at (1/n, 1/n) and of radius 1/(2n), wend that limn ‖fn‖w = 2 and limn ‖T (s, t)fn‖w = w(s, t), whence 1

2w(s, t) ≤ ‖T (s, t)‖follows ((s, t) ∈ R2

+). This implies that T (s, t) has a regular norm-function with respect

to w. We can see from the spectral radius formula that r(T (s, 0)) = es.

16

2.3 Spectra of representations


Let S], S∗ denote the characters and bounded characters of S, respectively. We note

that for any bounded character χ the inequality |χ(s)| ≤ 1 holds for every s ∈ S. We

recall that Cc(S) stands for the set of continuous functions with compact support in

S. The Fourier transform of a function f ∈ Cc(S) with respect to the representation

ρ : S → L(X) is given by

f(ρ) :=∫Sf(s)ρ(s) dµ(s).

The integral exists pointwise: f(ρ)x =∫S f(s)ρ(s)x dµ(s) (x ∈ X) in the Bochner sense

(see e.g. [27, Chapter 7.5]). Since the representation ρ is strongly continuous, the

Uniform Boundedness Principle implies that ‖ρ(s)‖ is bounded on the support of f ,

hence f(ρ) ∈ L(X). The characters of S are one-dimensional representations, hence the

formula for f(χ) also makes sense when χ ∈ S].Since S is a locally compact Hausdor space it is not hard to see by the Urysohn

lemma that for each χ ∈ S] there exists an f ∈ Cc(S) such that f(χ) 6= 0. Moreover,

for all distinct χ1, χ2 ∈ S] we can nd an f ∈ Cc(S) such that f(χ1) 6= f(χ2). In other

words the functions f separate the points of S] from each other and from zero.

We shall dene the spectrum for unbounded representations related to Lyubich's

δ-spectrum [49] and Kérchy's algebraic and balanced spectra [37]. In the discrete case

the algebraic spectrum of the representation ρ was dened in the following way (see

[37]). Let Aρ denote the Banach algebra generated by the set ρ(s) : s ∈ S, thenσa(ρ) := h ρ : h ∈ Σ(Aρ), where Σ(Aρ) is the Gelfand spectrum of the commutative

Banach algebra Aρ. It is easy to see that σa(ρ) is equal to the set of characters χ ∈ S]

which satisfy the condition∣∣∑

s∈F asχ(s)∣∣ ≤ ∥∥∑s∈F asρ(s)

∥∥ for all nite subsets F of S

and as ∈ C. In the continuous case one is led to the following extensions of the concept

of algebraic and balanced spectra.

Denition 2.3.1. The algebraic spectrum of the representation ρ is

σa(ρ) :=χ ∈ S] : |f(χ)| ≤ ‖f(ρ)‖ for all f ∈ Cc(S)

.

The balanced spectrum is dened by

σb(ρ) := σa(ρ) ∩ S]b,

17


where S]b := χ ∈ S] : χ(s) 6= 0 for all s ∈ S.The spectrum of ρ with regular norm-function is

σ(ρ) := χ ∈ σa(ρ) : |χ| ≤ cρ,

where cρ denotes the limit functional of ρ.

Now we are going to show that the balanced spectrum σb(ρ) is contained in the

spectrum σ(ρ). To do this, we need a lemma.

Lemma 2.3.2. If the norm-function of the representation ρ : S → L(X) is continuous

at s0 ∈ S, then |χ(s0)| ≤ ‖ρ(s0)‖ holds for every χ ∈ σa(ρ).

Proof. Let us take a character χ ∈ σa(ρ) and a positive ε > 0. Since the functions ‖ρ(s)‖and χ(s) are continuous at s0, there exists an open set V ⊆ S containing s0 such that

|‖ρ(s)‖− ‖ρ(s0)‖| < ε and |χ(s)− χ(s0)| < ε are valid for every s ∈ V. By the Urysohn

lemma we can nd a positive function f0 ∈ Cc(S) such that its support is contained in

V and∫S f0 dµ = 1. Then the inequalities

‖f0(ρ)x‖ ≤∫Vf0(s)‖ρ(s)x‖ dµ(s) ≤

∫Vf0(s)(‖ρ(s0)‖+ ε)‖x‖ dµ(s)

= (‖ρ(s0)‖+ ε)‖x‖ (x ∈ X)

imply ‖f0(ρ)‖ ≤ ‖ρ(s0)‖+ ε. Taking into account the fact that |f0(χ)− χ(s0)| < ε also

holds, we may infer that

|χ(s0)| − ε < |f0(χ)| ≤ ‖f0(ρ)‖ ≤ ‖ρ(s0)‖+ ε.

Since ε can be arbitrarily small, we conclude that |χ(s0)| ≤ ‖ρ(s0)‖.

Proposition 2.3.3. If ρ : S → L(X) is a representation with regular norm-function,

then σb(ρ) ⊆ σ(ρ).

Proof. Let Sc(ρ) stand for the set of points s ∈ S where ‖ρ(s)‖ is continuous. The

norm-function ‖ρ(s)‖, being the least upper bound of the continuous functions ‖ρ(s)x‖(x ∈ X, ‖x‖ = 1), is lower semicontinuous. It follows that S\Sc is of rst Baire category,and so Sc(ρ) is dense in S. (See e.g. [21, p. 87].)

Let us take a character χ ∈ σb(ρ). We know from Lemma 2.3.2 that |χ(s)| ≤ ‖ρ(s)‖for every s ∈ Sc(ρ). Let us consider the representation ρ := χ−1ρ. It is clear that

‖ρ(s)‖ ≥ 1 for every s ∈ Sc(ρ). Let us assume that ‖ρ(s0)‖ < 1 holds for some s0 ∈ S.Taking an open set V in S with compact closure, the number α := sup‖ρ(s)‖ : s ∈ V

18


is nite. For any n ∈ N and any s ∈ V, we have ‖ρ(ns0 + s)‖ ≤ ‖ρ(s0)‖nα. If n0 ∈ Nis suciently large, then ‖ρ(s0)‖n0α < 1, and so ‖ρ(n0s0 + s)‖ < 1 is true for every

s ∈ V. However, since (n0s0 + V ) ∩ Sc(ρ) is non-empty we arrive at a contradiction.

Thus ‖ρ(s)‖ ≥ 1 and so |χ(s)| ≤ ‖ρ(s)‖ holds for all s ∈ S.Assume ρ has p-regular norm function. Since |χ(s)| ≤ ‖ρ(s)‖ ≤ p(s) (s ∈ S), we

have that p(s) ≥ 1 (s ∈ S) is true for the gauge function p = |χ|−1p. Lemma 2.2.2

tells us that cp ≥ 1, and the relations cp = |χ|−1cp = |χ|−1cρ imply |χ| ≤ cρ, and so

χ ∈ σ(ρ).

Remark 2.3.4. For any χ ∈ σb(ρ) we know that |χ(s)|n = |χ(ns)| ≤ ‖ρ(ns)‖ ≤‖ρ(s)n‖ (s ∈ S, n ∈ N), whence it follows that |χ(s)| ≤ r(ρ(s)), with an arbitrary

representation ρ.

Remark 2.3.5. If S = Rn+, then it can be easily veried that every character χ ∈ (Rn

+)]

is non-vanishing, thus σa(ρ), σb(ρ) and σ(ρ) coincide in this case. (Indeed, let eini=1

be the standard basis in Rn+. If χ(t) = 0 holds for some t = (t1, . . . , tn) ∈ Rn

+, then the

equation χ(t) =∏ni=1 χ(ei)ti tells us that χ(ej) = 0 must be true for some 1 ≤ j ≤ n.

Hence χ vanishes on a dense set. Since χ(0) = 1, it follows that χ is not continuous.)

Remark 2.3.6. Let T be a bounded, linear operator on X. We denote the representa-

tion induced by T by ρT : Z+ → L(X). Then it can be shown that σa(ρT ) = σb(ρT ) =σ(T ), where σ(T ) denotes the polynomially convex hull of the spectrum of T [55, The-

orem 2.10.3].

Actually, a geometrically similar result can be veried for C0-semigroups as well; for

its proof we refer the reader to Proposition 3.2.3.

The most important spectrum that we shall need in the following is the peripheral

spectrum.

Denition 2.3.7. The peripheral spectrum of the representation ρ : S → L(X) with

regular norm-function is dened by

σper(ρ) := χ ∈ σ(ρ) : |χ(s)| = cρ(s) for all s ∈ S .

Next, the point spectrum will be dened.

Denition 2.3.8. The point spectrum of the representation ρ : S → L(X) is

σp(ρ) := χ ∈ S] : there exists a 0 6= x ∈ X with ρ(s)x = χ(s)x for all s ∈ S.

19


The adjoint ρ∗(s) := ρ(s)∗ (s ∈ S) of ρ is not necessarily strongly continuous, hence

the spectrum of ρ∗ cannot be dened in general. However, there is no diculty dening

σp(ρ∗) in an analogous way to σp(ρ).

The multiplication of the representation ρ (of regular norm-function) by a non-

vanishing character τ yields a representation of a regular norm-function whose various

kinds of spectra can be naturally derived from the spectra of ρ.

Lemma 2.3.9. If τ ∈ S]b then σa(τρ) = τσa(ρ), σb(τρ) = τσb(ρ), σ(τρ) = τσ(ρ),σper(τρ) = τσper(ρ), σp(τρ) = τσp(ρ) and σp(τρ∗) = τσp(ρ∗).

Proof. Consider the bijection Ψ: Cc(S)→ Cc(S), Ψ(f) := fτ. Since (Ψ(f))(χ) = f(τχ)and (Ψ(f))(ρ) = f(τρ) hold for every f ∈ Cc(S), we may infer that σa(τρ) = τσa(ρ),whence σb(τρ) = τσb(ρ) readily follows. In view of cτρ = τcρ we deduce that σ(τρ) =τσ(ρ) and σper(τρ) = τσper(ρ). Finally, the equalities σp(τρ) = τσp(ρ) and σp(τρ∗) =τσp(ρ∗) are immediate consequences of the denition.

Let us equip S] with the compact-open topology, so that convergence of a net in

S] means uniform convergence on compact subsets of S. The closed set S∗ of bounded

characters can be identied with the Gelfand spectrum Σ(L1(S)) of the abelian Banach

algebra L1(S). Indeed, L1(S) with the multiplication (f ∗ g)(s) =∫S f(s− t)g(t) dµ(t)

is a closed subalgebra of the abelian Banach algebra L1(G). Adapting the proof of [58,

Theorem 1.2.2] to a semigroup setting tells us that the mapping

Λ: S∗ → Σ(L1(S)), χ 7→ hχ, where hχ(f) =∫Sfχ = f(χ),

is a bijection. We recall that Σ(L1(S)) is a locally compact Hausdor space with

the Gelfand topology, induced by the weak-∗ topology of the dual space of L1(S).

Notice also that if f ∈ C0(Σ(L1(S))) is the Gelfand transform of f ∈ L1(S), then

f(hχ) = hχ(f) = f(χ) holds for every χ ∈ S∗.

Proposition 2.3.10. The mapping Λ: S∗ → Σ(L1(S)), χ 7→ hχ is a homeomorphism,

and so S∗ is locally compact in the compact-open topology.

Proof. For the sake of completeness we shall sketch a proof. The continuity of Λ is an

immediate consequence of the fact that Cc(S) is dense in L1(S) and that ‖h‖ = 1 is

true for every h ∈ Σ(L1(S)).

20


Let us now assume that the net hν = hχνν∈N converges to h = hχ; that is,

limν hν(g) = h(g) holds for every g ∈ L1(S). Let us x a function f ∈ L1(S) such that

h(f) 6= 0, and let us consider the equations

hν(f)χν(t) = hν(f−t) (ν ∈ N, t ∈ S)

and

h(f)χ(t) = h(f−t) (t ∈ S).

(For their validity we refer to the proof given in [58, Theorem 1.2.2].) Since the mapping

ϕ : S → L1(S), t 7→ f−t is continuous, we can readily deduce that χνν converges to χuniformly on compact sets.

Taking any τ ∈ S]b, the multiplication Mτ : S] → S], χ 7→ τχ is clearly a homeo-

morphism, and so τS∗ is also a locally compact Hausdor space with the compact-open

topology.

Now we shall turn to the question of the spectrum of a representation.

Proposition 2.3.11. Let ρ : S → L(X) be a representation of regular norm-behaviour.

The spectrum σ(ρ) of ρ is a locally compact Hausdor space with the compact-open

topology, and so is its closed subset σper(ρ).

Proof. It is not hard to see that the algebraic spectrum σa(ρ) is closed in S]. As the

limit functional cρ belongs to S]b, it follows from Proposition 2.3.10 that cρS∗ is locally

compact. Then the closed subset σ(ρ) = σa(ρ)∩cρS∗ of cρS∗ is also locally compact.

Remark 2.3.12. Let us assume that the representation ρ : S → L(X) is bounded:

α := sup‖ρ(s)‖ : s ∈ S < ∞. If ‖ρ(s0)‖ < 1 holds for some s0 ∈ S, then the

inequalities ‖ρ(ns0 + s)‖ ≤ ‖ρ(s0)‖nα (n ∈ N) show that lims ‖ρ(s)‖ = 0, i.e. ρ is

uniformly stable.

Assuming that ‖ρ(s)‖ ≥ 1 is true for every s ∈ S, we can see that ρ is of regular normbehaviour with respect to the gauge function p(s) := α (s ∈ S). The limit functional cρof ρ is clearly the constant 1 function. Thus σper(ρ) coincides with the unitary spectrum

σu(ρ) := χ ∈ σ(ρ) : |χ| = 1 of ρ. It can be easily veried that σa(ρ) = σ(ρ) is also

true. Indeed, if |χ(s0)| > 1 holds for some χ ∈ S] and s0 ∈ S, then limn |χ(ns0)| =∞,and by the Urysohn lemma we nd that sup|f(χ)| : 0 ≤ f ∈ Cc(S), ‖f‖1 = 1 = ∞.Since sup‖f(ρ)‖ : 0 ≤ f ∈ Cc(S), ‖f‖1 = 1 ≤ α, we obtain that χ cannot belong to

σa(ρ).The Fourier transforms f(χ) and f(ρ) are clearly dened for every f ∈ L1(S) and

χ ∈ S∗. Taking into account the fact that Cc(S) forms a dense subset of L1(S), we

21

2.4 Regularity and isometric representations

may conclude that if |f(χ)| ≤ ‖f(ρ)‖ holds for every f ∈ Cc(S) then so does follow for

every f ∈ L1(S). Thus σ(ρ) coincides with the spectrum introduced by Batty and Vu

for bounded representations in [8]. We recall also that this concept is an adaptation of

the nite L-spectrum and the Arveson spectrum, dened for group representations, to

a semigroup setting (see [49] and [15]).

Remark 2.3.13. Let ρ : S → L(X) be a representation of regular norm-behaviour.

Since cρ ∈ S]b, the representation ρ := c−1ρ ρ : S → L(X) is also of regular norm-behaviour

and cρ = 1. Clearly |χ| = 1 holds for the characters in σper(ρ). Hence every χ ∈ σper(ρ)can be uniquely extended to a character χ of the extension group G. We conclude that

σper(ρ) can be identied with the unitary spectrum Spu(ρ) introduced in [6], namely

σper(ρ) = χ|S : χ ∈ Spu(ρ). Therefore σper(ρ) = cρ(χ|S) : χ ∈ Spu(ρ) is true by

Lemma 2.3.9.


The key ingredient in proving the stability of ρ is the transmission of the conditions

to a related isometric representation. The method is well known for bounded represen-

tations and we can extend it using the regularity condition. This kind of associated

representation is included in the following theorem.

Theorem 2.4.1. For any representation ρ : S → L(X) with p-regular norm-function,

there exists an isometric representation ψ : S → L(Y) on a Banach space Y and a

contractive transformation Q ∈ L(X,Y) such that:

(i) ker Q = x ∈ X : a- lims ‖ρ(s)x‖/p(s) = 0, and ranQ is dense in Y,

(ii) Qρ(s) = cρ(s) ψ(s)Q holds for every s ∈ S,

(iii) for every operator C ∈ ρ(S)′, there exists a unique operator D ∈ ψ(S)′ suchthat QC = DQ; furthermore, the mapping γ : ρ(S)′ 7→ ψ(S)′, C 7→ D is a

contractive algebra-homomorphism,

(iv) σ(ρ) ⊇ cρσ(ψ), σper(ρ) ⊇ cρσper(ψ), σp(ρ∗) ⊇ cρσp(ψ∗).

Proof. (i): Dene a seminorm on X in the following way:

`(x) := supm

(‖ρ(·)x‖

p

): m ∈M(S)

.

22


One sees immediately that `(x) ≤ ‖x‖, and `(x) = 0 if and only if a-lims ‖ρ(s)x‖/p(s) =0. Let Y be the completion of the quotient space X/ker ` with norm ‖x+ ker `‖ := `(x),and let Q : X→ Y denote the natural embedding. Obviously, ran Q is dense.

(ii): Applying Lemma 2.1.13, for any x ∈ X and t ∈ S we have

‖Qρ(t)x‖ = `(ρ(t) x) = supm∈M(S)

m

(‖ρ(·+ t)x‖

p

)= sup

m∈M(S)m

(‖ρ(·+ t)x‖

pt

ptp

)= cρ(t) sup

m∈M(S)m

(‖ρ(·)x‖

p

)= cρ(t) `(x) = cρ(t) ‖Qx‖.

Hence there exists a unique isometry ψ(t) ∈ L(Y) such that Qρ(t) = cρ(t) ψ(t)Q. Thisintertwining relation readily yields that ψ(s+ t) = ψ(s)ψ(t), so ψ is a homomorphism.

The inequality

‖cρ(s)ψ(s)Qx− cρ(t)ψ(t)Qx‖ = ‖Qρ(s)x−Qρ(t)x‖

≤ ‖Q‖‖ρ(s)x− ρ(t)x‖ (x ∈ X)

shows that the function cρ(t)ψ(t)y is continuous in t for every y in the dense range of

Q, and then so is the function ψ(t)y. Taking into account the fact that ψ is isometric,

we may infer that it is strongly continuous. Thus ψ is an isometric representation.

(iii): Taking any C ∈ ρ(S)′, we have that

‖QCx‖ = supm∈M(S)

m

(‖Cρ(·)x‖

p

)≤ ‖C‖`(x) = ‖C‖‖Qx‖,

so there exists a unique D ∈ L(Y) such that QC = DQ. Clearly ‖D‖ ≤ ‖C‖, and it is

easy to check that D ∈ ψ(S)′ and that γ is an algebra homomorphism.

(iv): Since f(ρ) ∈ ρ(S)′ and Qf(ρ) = f(cρψ)Q, we have γ(f(ρ)) = f(cρψ) (f ∈Cc(S)). Recalling the fact that the mapping γ : ρ(S)′ → ψ(S)′ is contractive by

(iii), we may infer that ‖f(cρψ)‖ ≤ ‖f(ρ)‖, whence cρσa(ψ) = σa(cρψ) ⊆ σa(ρ) follows(see Lemma 2.3.9). As the limit functional of the isometric representation ψ is clearly

the constant 1 function, we conclude that cρσper(ψ) ⊆ σper(ρ). Finally, the equations

ker Q∗ = 0 and ρ∗(s)Q∗ = cρ(s) Q∗ψ∗(s) (s ∈ S) readily imply that σp(ρ∗) ⊇cρσp(ψ∗).

Now we will deduce the following corollary.

23

2.5 The stability theorem

Corollary 2.4.2. If ρ : S → L(X) is a representation with p-regular norm-function and

a- lims ‖ρ(s)x‖/p(s) = 0 does not hold for some x ∈ X, then σper(ρ) is not empty.

Proof. The conditions ensure that the associated isometric representation ψ of Theorem

2.4.1 acts on a non-zero Banach space Y. Thus σper(ψ) is non-empty by Corollary 3.3

in [8]. Since cρσper(ψ) is included in σper(ρ), we infer that σper(ρ) is not empty.

2.5 The stability theorem

Now we are in a position to prove a stability result in terms of almost convergence for

representations with a regular norm-function.

Theorem 2.5.1. Let ρ : S → L(X) be a representation with a p-regular norm-function.

If σper(ρ) is countable and σp(ρ∗) ∩ χ : |χ| = cρ is empty then

a- lims

‖ρ(s)x‖p(s)

= 0

holds for all x ∈ X.

Proof. Supposing it is not the case, then by Theorem 2.4.1 the associated isometric

representation ψ : S → L(Y) acts on a non-zero Banach space Y. Hence, by Corollary

3.3 in [8], σper(ψ) is not empty. The relation σper(ρ) ⊇ cρσper(ψ) implies that σper(ψ) isalso countable. Thus σper(ψ) contains, by Proposition 4.1 in [8], an eigenvalue χ of ψ∗.

Applying Theorem 2.4.1 again, we conclude that cρχ must belong to σp(ρ∗) ∩ σper(ρ),which is a contradiction.

Applying Proposition 2.1.10 we obtain the following corollary.

Corollary 2.5.2. Let ρ : S → L(X) be a representation with a p-regular norm-function.

If σper(ρ) is countable and σp(ρ∗) ∩ χ : |χ| = cρ is empty then

limi→∞

1µ(Ki)

∫Ki

‖ρ(s)x‖p(s)

dµ(s) = 0

is true for all x ∈ X, where Kii is any Følner sequence.

In view of Proposition 2.2.4 and the Remark 2.3.12, Theorem 2.5.1 is a generalization

of the stability result Theorem 1.1.2 for bounded representations. From Remark 2.3.13,

the spectral conditions of Theorem 2.5.1 are essentially the same as those in the main

result of Theorem 3.2 in [6]. The dierences lie in the norm-condition on ρ and in the

nature of convergence expressing stability.

24

Chapter 3

C0-semigroups and regularity

In this chapter we shall introduce a slightly dierent notion of regularity from that

presented in Chapter 2. Our aim is to characterize this kind of regularity for continuous

one-parameter operator semigroups. To do this, rst we will outline some basic prop-

erties of topologically invariant means which are known from the theory of amenable

groups. Then, after making some preliminary remarks, we will dene the class of C0-

semigroups with topologically regular norm-function. It turns out that this type of

regularity can be described in terms of an integral condition. Next we will introduce

the regularity constant which plays a crucial role in the characterization. We will show

that topological regularity of a representation is equivalent to the positivity of the reg-

ularity constant.

3.1 Topologically invariant means

Let R+ denote the additive semigroup of nonnegative real numbers. We use the notation

L∞(R+) for the Banach space of essentially bounded, Lebesgue measurable functions

with the usual ‖ · ‖∞ norm. We recall that a mean m is a continuous linear functional

on L∞(R+) which satises ‖m‖ = m(1) = 1 (here 1 denotes the constant 1 function on

R+). We say that a mean m is an invariant mean if m(f) = m(fs) for any f ∈ L∞(R+)

and s ∈ R+, where the function fs : R+ → C is dened by fs(t) := f(s + t) (t ∈ R+).

Let M(R+) stand for the set of all invariant means. Furthermore, let G denote the set of

non-negative, measurable functions g on R+ which satisfy the condition∫∞

0 g(s)ds = 1.

25


For any f ∈ L∞(R+) and g ∈ G, let us consider the convolution f ∗g ∈ L∞(R+), dened

by (f ∗ g)(y) =∫∞

0 f(s+ y)g(s) ds.

Denition 3.1.1. We say that a mean m is topologically invariant if m(f ∗ g) = m(f)holds for every f ∈ L∞(R+) and any g ∈ G.

These kinds of means were originally introduced by A. Hulanicki for locally compact

Hausdor groups (see [56, p. 9]). For convenience, we shall use the notation Mt for the

set of topologically invariant means. In our rst statement we will show that Mt(R+)

is not empty. This is well-known for groups and we will simply reproduce the proof in

the semigroup context for the sake of completeness.

Proposition 3.1.2. Mt(R+) is not empty.

Proof. Let us take the functionals ϕn(f) := 1n

∫ n0 f(s) ds (n ∈ N) on L∞(R+). Then it

follows that ‖ϕn‖ = ϕn(1) = 1 for any n ∈ N. Since the unit ball of the dual space

L∞(R+)∗ is weak-∗ compact, the sequence ϕnn has a weak-∗ cluster point. We will

now show that the cluster points are topologically invariant means. Let us have an

f ∈ L∞(R+) and g ∈ G. Choosing some ε ∈ (0, 1), we can nd a t0 ∈ R+ such that∫∞t0g(s) ds < ε. Since

ϕn(f ∗ g)− ϕn(f) =1n

∫ n

0

(∫ ∞0

(f(s+ t)g(t)− f(s)g(t)) dt)ds

=∫ ∞

0

(1n

∫ n

0(f(s+ t)g(t)− f(s)g(t)) ds

)dt

=∫ t0

0

1n

∫[t,t+n]\[0,n]

f(s) ds−∫

[0,n]\[t,t+n]

f(s) ds

g(t) dt

+∫ ∞t0

(1n

∫ n

0(f(s+ t)− f(s)) ds

)g(t) dt,

we may infer that

|ϕn(f ∗ g)− ϕn(f)| ≤ ‖f‖∞ · |[0, n]4[t0, t0 + n]| · n−1

∫ t0

0g(t) dt

+2‖f‖∞∫ ∞t0

g(t) dt

≤ ‖f‖∞2t0n−1 + 2‖f‖∞ε ≤ 3‖f‖∞ε,

26


provided n is suciently large (here |A| denotes the Lebesgue measure of a measurable

set A ⊆ R+). Since ε > 0 was arbitrary, we have limn |ϕn(f ∗ g) − ϕn(f)| = 0. Thusany weak-∗ cluster point of ϕnn is a topologically invariant mean.

A little reasoning shows that every topologically invariant mean m is a (translation)

invariant mean. Indeed, for any xed f ∈ L∞(R+) and y ∈ R+, let us choose a function

g ∈ G such that the support of g is included in y + R+. We have

m(f) = m(f ∗ g) = m

(∫ ∞y

f( ·+ s)g(s) ds)

= m(fy ∗ gy) = m(fy)

because gy is in G.

In the following we shall construct a function f0 ∈ L∞(R+) such that every topolog-

ically invariant mean vanishes on f0, but m(f0) = 1 holds with an appropriate invariant

mean m. Thus ∅ 6= Mt(R+), Mt(R+) ⊂ M(R+) and Mt(R+) 6= M(R+). First ob-

serve that if f ∈ L∞(R+) ∩ L1(R+), then m(f) = 0 holds for every m ∈ Mt(R+).

Indeed, denoting the characteristic function of [0, n] by χn, the asymptotic relation

limn ‖f ∗ 1nχn‖∞ = 0 implies that limnm(f ∗ 1

nχn) = 0 holds for every mean m. If

m ∈ Mt(R+) then m(f ∗ 1nχn) = m(f) (n ∈ N), and so m(f) = 0. Let r1, r2, . . . be an

enumeration of the nonnegative rational numbers. Setting Ω =⋃∞n=1(rn−2−n, rn+2−n),

let f0 be the characteristic function of Ω ∩ R+. Since f0 ∈ L∞(R+) ∩ L1(R+), we get

that m(f0) = 0 for every m ∈Mt(R+).

The following statement is valid as well.

Proposition 3.1.3. There exists an invariant mean m on L∞(R+) such that m(f0) = 1for the function f0 dened above.

Proof. Let W denote the linear span of the functions (f0)s (s ∈ R+) and the constant

1. Let m0 be the linear functional on W , dened by m0(f) :=∑n

k=0 αk whenever

f = α0 · 1 +∑n

k=1 αk(f0)sk (αk ∈ C, sk ∈ R+, n ∈ N). Since (f0)sk is the characteristic

function of the dense, open set Ωk := (Ω − sk) ∩ R+ in R+, and since⋂nk=1 Ωk is also

a dense open subset of R+, we may infer that |m0(f)| ≤ ‖f‖∞. Noting the linearity

property, we nd thatm0 is well-dened. Furthermore,W andm0 are clearly translation

invariant.

Now let N denote the set of all norm-preserving extensions ofm0 fromW to L∞(R+).Then N is a non-empty, convex, weak-∗ compact set in the dual space L∞(R+)∗. Forany s ∈ R+, let T (s) : N→ N be dened by (T (s)m)(f) := m(fs) (f ∈ L∞(R+)). Since

27


the commuting mappings T (s) (s ∈ R+) are ane and weak-∗ continuous, the Markov

Kakutani theorem implies the existence of a common xed point m ∈ N. Whence it

follows that m is an invariant mean and m(f0) = m0(f0) = 1.

Generally, if G is a non-compact, non-discrete and locally compact group which is

amenable, the sets Mt(G) and M(G) are dierent. For details, we suggest [56, p. 277],

see also [26], [59], [62].

For any compact set K ⊆ R+ of positive measure, we introduce the mean ϕK on

L∞(R+), dened by ϕK(f) := 1|K|∫K f(s)ds.We recall that a sequence of compact sets

Knn ⊆ R+ with non-empty interiors is a Følner sequence if

|(s+Kn) 4 Kn|/|Kn| → 0 (n→∞)

uniformly in s ∈ K, for any compact set K ⊆ R+. The following statement describes

the set Mt(R+). Its counterpart for groups is due to C. Chou. We sketch a proof, which

follows the lines of [56, p. 138], for the sake of completeness.

Theorem 3.1.4. The set Mt(R+) is the weak-∗ closure of the convex hull of the set of

all weak-∗ cluster points of the sequences

ϕKn+snn∈N (snn ∈ RN+),

where Knn is an arbitrarily given xed Følner sequence in R+.

Proof. We can see from the proof of Proposition 3.1.2 that every such weak-∗ clusterpoint is a topologically invariant mean, and then so are the means in the weak-∗ closureof the convex hull.

To prove the converse statement, let us denote the weak-∗ closure of the convex hull

of the above cluster points by Lt. Suppose that there is a functional φ ∈ Mt(R+) \ Lt.

By the HahnBanach theorem there exists a function f ∈ L∞(R+), a real number c andan ε > 0 such that

Re m(f) ≤ c < c+ ε ≤ Re φ(f) for all m ∈ Lt.

Every mean is a positive functional which implies Re m(f) = m(Re f) and Re φ(f)= φ(Re f), thus we may suppose that f is real-valued. Since the functional φ is topo-

logically invariant, and since φ is monotone increasing as a mean, it follows that

φ(f) =1|Kn|

φ(f ∗ χKn) ≤ sups

1|Kn|

∫Kn

f(s+ t) dt ≤ ‖f‖∞ <∞.

28


For every n ∈ N choose an sn ∈ R+ such that

sups

1|Kn|

∫Kn

f(s+ t) dt− 1|Kn|

∫Kn

f(sn + t) dt < ε/2.

Let m0 denote a weak-∗ cluster point of the sequence ϕKn+sn, where clearly m0 ∈ Lt.

Thus, for some large n, |ϕKn+sn(f) −m0(f)| < ε/2, and so the following inequalities

are satised

1|Kn|

∫Kn+sn

f(t) dt < c+ ε/2 ≤ φ(f)− ε/2 ≤ 1|Kn|

∫Kn

f(sn + t) dt.

This is a contradiction, hence the theorem must be valid.

Denition 3.1.5. A function f ∈ L∞(R+) is called topologically almost convergent if

the set m(f) : m ∈ Mt(R+) is a singleton. We shall use the notation at-limf = c

whenever m(f) = c for all m ∈Mt(R+).

Denition 3.1.6. We say that a function f ∈ L∞(R+) is topologically almost convergent

in the strong sense to a complex number c if at-lim|f − c| = 0.

We note that at-lim|f − c| = 0 if and only if m(fg) = cm(g) for every g ∈ L∞(R+)

and m ∈Mt(R+). The proof follows the same argument as the proof of Lemma 2.1.13,

which played an important role in the previous chapter.

The following characterization will be very useful for us. (Notice that its counterpart

in `∞(Z+) is the classical characterization of almost convergent sequences by Lorentz

[49].)

Proposition 3.1.7. An f ∈ L∞(R+) is topologically almost convergent to c if and only

if

limn→∞

1|Kn|

∫Kn

fy(s) ds = c

uniformly with respect to y ∈ R+, where Knn is an arbitrarily chosen Følner sequence.

Proof. Let us assume that there exists an ε > 0, a strictly increasing sequence nkk ofpositive integers, and a sequence ykk ∈ R+ such that∣∣∣∣∣ 1

|Knk |

∫Knk

fyk(s) ds− c

∣∣∣∣∣ > ε

29


is true for every k ∈ N. Then Ck := Knk + ykk is also a Følner sequence. For every

k ∈ N, let us consider the linear functional ϕk on L∞(R+) dened by

ϕk(h) :=1|Ck|

∫Ck

h(s) ds , h ∈ L∞(R+).

We know from Theorem 3.1.4 that any weak-∗ cluster point m0 of the sequence ϕkkis in Mt(R+). Since |m0(f) − c| ≥ ε by our assumption, we conclude that f cannot

topologically almost converge to c. This proves the necessity of the integral condition.

The converse is a direct corollary of Theorem 3.1.4.

We recall from Section 2.1 that a function f ∈ L∞(R+) almost converges to c ∈ C,that is, a-limf = c, ifm(f) = c for all invariant meansm ∈M. It is clear that a-limf = c

implies at-limf = c, but the converse does not follow (see Proposition 3.1.3). The

advantage of considering topologically almost convergent functions instead of almost

convergent functions appears in the complete characterization given in Proposition 3.1.7.

Remark 3.1.8. We note that the results of this section remain true in a more general

framework. Let us consider a locally compact, σ-compact, Hausdor abelian group

(G,+), and let (S,+) be a closed subsemigroup of G with non-empty interior S such

that S − S = G and S ∩ (−S) = 0. Since these groups are amenable, we can nd a

Følner sequence in G, and even in S (see [56, Theorem 4.16] and [42]). Chou's theorem

works in this situation as well, hence Mt(S) 6= ∅ and the above integral characterization

is also true.

3.1.1 An immediate application

In the following, we will give an application of (topologically) almost convergent func-

tions.

Let X be a complex Banach space and pick an x ∈ X. Let us form the closure of

the subspace spanned by the vectors T (s)x : s ∈ R+, denoted by Xx, where T is a

bounded C0-semigroup. Obviously, Xx is T -invariant hence the restriction of T to Xx,

Tx := T|Xx is a well dened C0-semigroup whose generator is denoted by Ax.

Theorem 3.1.9. Let T : R+ → L(X) be a bounded C0-semigroup with the generator A.

Then the following assertions are equivalent for any x ∈ X :

(i) σp(A∗x) ∩ iR = ∅,

(ii) 〈x∗, eiλsT (s)x〉 topologically almost converges to zero for every λ ∈ R and x∗ ∈ X∗x,

30


(iii) 〈x∗, eiλs T (s)x〉 almost converges to zero for every λ ∈ R and x∗ ∈ X∗x,

(iv) inf∥∥∥∥∫

R+

eiλsg(s)T (s)x ds∥∥∥∥ : g ∈ L1(R+),

∫R+

g = 1

= 0 for every λ ∈ R.

Proof. (i)⇒ (ii): Let us assume that (ii) is not satised. Then there exists an x∗ ∈ X∗x,

λ ∈ R and m ∈Mt(R+) such that m(〈x∗, eiλsT (s)x〉) 6= 0. Using Theorem 3.1.4 we can

nd a sequence sn∞n=1 ∈ RN+ and a 0 6= c ∈ C such that

limn→∞

1n

∫ sn+n

sn

〈x∗, eiλsT (s)x〉 ds = c.

Let us dene for each n ∈ N the linear functional z∗n on Xx where

〈z∗n, y〉 :=1n

∫ sn+n

sn

〈x∗, eiλsT (s)y〉 ds, y ∈ Xx.

Since supn ‖z∗n‖ < ∞, the BanachAlaoglu theorem implies the existence of a weak-∗cluster point of the sequence z∗n∞n=1 denoted by z∗. We nd that z∗ is non-zero on Xx

because 〈z∗, x〉 = c and T ∗x (t)z∗ = e−iλsz∗ holds for every t ∈ R+. Indeed, if y ∈ Xx,

|〈z∗n, eiλt T (t)y − y〉| =∣∣∣∣ 1n∫ sn+n

sn

〈x∗, eiλ(s+t)T (s+ t)y − eiλsT (s)y〉 ds∣∣∣∣

≤ ‖x∗‖ sups‖eiλsT (s)y‖|(t+ sn + [0, n])4(sn + [0, n])|

n

→ 0 (n→∞),

which implies T ∗x (t)z∗ = e−iλtz∗ for every t ∈ R+. However, this means that A∗xz∗ =

−iλz∗, which is a contradiction, so the implication is true.

(ii)⇒ (i): Let us choose an iλ ∈ σp(A∗x) with corresponding eigenvector 0 6= x∗ ∈ D(A∗x).Then

e−iλtT ∗x (t)x∗ = x∗ for all t ∈ R+(3.1)

holds (see [20, Chapter II.2 and Theorem IV.3.6]) and there exists an s ∈ R+ with the

property 〈x∗, T (s) x〉 6= 0. For any m ∈Mt(R+) ⊆M(R+), we have

0 = m(〈x∗, e−iλtT (t)x〉) = m(〈x∗, e−iλ(t+s)T (t+ s)x〉)

= e−iλs m(〈e−iλtT ∗x (t)x∗, T (s)x〉).

But from (3.1) we have m(〈e−iλtT ∗x (t)x∗, T (s)x〉) = m(〈x∗, T (s) x〉 · 1) 6= 0. This is acontradiction, so σp(A∗x) ∩ iR = ∅.(ii)⇔ (iii) We can easily see that the function ψλ(s) := eiλsT (s)x (λ ∈ R) is a boundeduniformly continuous X-valued function on R+, i.e. ψλ ∈ BUC(R+,X). On the other

31


hand, the left translation semigroup S(s) : BUC(R+,C)→ BUC(R+,C), (S(s)f)(t) :=f(t + s) (s, t ∈ R+) is a C0-semigroup on BUC(R+,C). Hence, for any m ∈ M(R+)and g ∈ G ⊂ L1(R+), we may infer that

m(f) =∫ ∞

0g(s)m(S(s)f) ds = m

(∫ ∞0

g(s) S(s)f ds)

(see [49, TheoremDenition p. 2]). This means that the set of the topologically in-

variant means and the set of the invariant means restricted to the closed subspace

BUC(R+,C) of L∞(R+) are the same. We conclude that m(〈x∗, ψλ〉) : m ∈Mt(R+) =m(〈x∗, ψλ〉) : m ∈M(R+) for any x∗ ∈ X∗x.

(ii) ⇒ (iv): Fix a λ ∈ R. By (ii), m(〈x∗, eiλtT (t)x〉) = 0 is valid for every m ∈Mt(R+)and x∗ ∈ X∗x. We can see from Theorem 3.1.7 that

limn→∞

⟨x∗,

1n

∫ n

0eiλsT (s)x ds

⟩= 0,

because [0, n]∞n=1 is a Følner sequence on R+. This implies that n−1∫ n

0 eiλsT (s)x dsconverges to 0 as well (see [44, II. Theorem 1.5]). Since n−1χ[0,n] ∈ G, the implication

(ii) ⇒ (iv) is veried.

(iv) ⇒ (ii): Choosing an m ∈ Mt(R+), we readily see that m(g ∗ f) = m(f) for any

g ∈ L1(R+) such that∫

R+g(t) dt = 1. Hence the following straightforward inequalities

hold, if x∗ ∈ X∗x, λ ∈ R and g ∈ L1(R+) :

|m(〈x∗, eiλsT (s)x〉)| =∣∣∣∣m(∫ ∞

0g(t)〈x∗, eiλ(s+t)T (s+ t)x〉 dt

)∣∣∣∣≤ ‖x∗‖ sup

s∈R+

∥∥∥∥T (s)∫ ∞

0eiλtg(t)T (t)x dt

∥∥∥∥≤ ‖x∗‖ sup

s∈R+

‖T (s)‖∥∥∥∥∫ ∞

0eiλtg(t)T (t)x dt

∥∥∥∥ .We conclude that m(〈x∗, eiλsT (s)x〉) = 0 for any m ∈ Mt(R+), thus 〈x∗, eiλsT (s)x〉topologically almost converges to 0.

Remark 3.1.10. The above result is partly well-known [5, Proposition 3.2]) and goes

back to classical operator ergodic results, see [44, Theorem II.1.3]; however, to the best

of our knowledge, the originality of this approach seems to be in the use of (topologically)

almost convergent functions.

32

3.2 C0-semigroups and topological regularity


We would like to introduce an upper bound for the norm-function of C0-semigroups that

is well behaved. To do this, we will introduce the following type of gauge functions.

Denition 3.2.1. We say that p : R+ → [1,∞) is a topological gauge function if (i) it

is measurable, (ii) it is locally bounded (i.e. bounded on compact sets) (iii) for every

compact subset K of R+, supp(t + s)/p(t) : t ∈ R+, s ∈ K < ∞, and (iv) for every

s ∈ R+, ps/p topologically almost converges in the strong sense to a positive real number

cp(s). The function cp is called the limit functional of the gauge function p. Here the

set of topological gauge functions shall be denoted by Pt.

Let X denote a complex Banach space and let L(X) stand for the set of all bounded

linear operators on X. Also, let T : R+ → L(X) be a strongly continuous, one-parameter

semigroup, that is a C0-semigroup.

Denition 3.2.2. The C0-semigroup T : R+ → L(X) has regular norm behaviour

with respect to the topological gauge function p or has a p-regular norm-function if (i)

‖T (s)‖ ≤ p(s) holds for every s ∈ R+, and (ii) at-lims‖T (s)‖/p(s) = 0 does not hold.

We note that the results in Chapter 2 remain valid with this denition of regularity.

In fact, the main ingredients in that chapter, e.g. the limit functional and the associated

isometric representation, can be introduced in a similar way using Mt(R+) instead

of M. It can be also shown that if T has a p-regular norm-function, p ∈ Pt, then

the corresponding limit functional cT = cp is equal to the spectral radius function

r(T (s)) of T (see the proof of Proposition 2.2.12). Furthermore, it is well known that

r(T (s)) = eω0(T )s (s ∈ R+), where ω0(T ) = lims→∞ (log ‖T (s)‖)/s is the exponential

growth bound of T (see [20, p. 251]).

Now to get a complete adaptation of the main result in Chapter 2, we shall identify

the spectrum of an arbitrary C0-semigroup T : R+ → L(X) with an appropriate hull of

the spectrum of the innitesimal generator A of T. We will assume that w0(T ) > −∞,

which is the case when T has regular norm-behaviour.

Let R]+ denote the set of all characters of R+, that is the set of all continuous

representations of R+ on C. We note that R]+ = χλ : λ ∈ C, where χλ(s) = eλs

(s ∈ R+). In the general semigroup setting the spectrum of T is dened as the set of

33


characters χ ∈ R]+ satisfying the condition∣∣∣∣∫ ∞

0f(s)χ(s) ds

∣∣∣∣ ≤ ‖f(T )‖

for every f ∈ Cc(R+), the set of continuous functions with compact support on R+.

Here

f(T )x :=∫ ∞

0f(s) T (s)x ds (x ∈ X)

is the Fourier transform of f with respect to T. With the special form of the characters

in R]+ we can identify the spectrum of T as a subset of C, namely

σ(T ) = λ ∈ C : |f(λ)| ≤ ‖f(T )‖ for every f ∈ Cc(R+),

where f(λ) =∫∞

0 eλsf(s)ds (λ ∈ C) is the Laplace transform of f. The function f being

analytic on C, the set σ(T ) is closed. Moreover, since the characters are non-vanishing,

Re λ ≤ ω0(T ) holds for every λ ∈ σ(T ) (see e.g. the proof of Proposition 2.3.3). The

peripheral spectrum of T is dened as

σper(T ) := λ ∈ σ(T ) : Re λ = ω0(T ).

Notice that these denitions correspond to the ones given in Chapter 2 for regular

representations of general semigroups.

Now let us consider the innitesimal generator A of T, where it is known that A is

a densely dened closed operator. Applying the equations

T (t)x− x = A

∫ t

0T (s)x ds (t ∈ R+, x ∈ X)

and

T (t)x− x =∫ t

0T (s)Ax ds (t ∈ R+, x ∈ D(A))

(see e.g. the proof of Theorem 34.4 in [45]), it is straightforward to show that

f(σ(A)) ⊆ σ(f(T )) for every f ∈ Cc(R+).

Thus the closed set σ(A) is contained in σ(T ). Next, let ρ∞(A) stand for the open

component of C \ σ(A) containing the right half-plane z ∈ C : Re z > ω0(T ).

34


Proposition 3.2.3. With the previous notation, we have

σ(T ) = C \ ρ∞(A).

Proof. Replacing T (s) by e−(ω0(T )+1)sT (s) (s ∈ R+), we may assume that ω0(T ) =−1. Let Ω be a component of C \ σ(A), other than ρ∞(A). If Ω is bounded, then

the Maximum Principle applied to f (f ∈ Cc(R+)) tells us that Ω ⊆ σ(T ). If Ω is

unbounded, then we can consider the bounded open set Ω0 = 1/z : z ∈ Ω and to

the function F (z) = f(1/z) (z ∈ Ω−0 \ 0). Applying the Maximum Principle to the

function gn(z)F (z), where gn(reiϕ) := n√reiϕ/n (r ∈ [0,∞), ϕ ∈ (0, 2π)), and letting n

tend to innity, we again obtain that Ω ⊆ σ(T ).Since (f ∗ g)(T ) = f(T )g(T ) holds for every f, g ∈ Cc(R+), the norm closure AT

of the Fourier transforms f(T ) : f ∈ Cc(R+) forms a commutative Banach algebra.

Fixing a point λ0 ∈ C with Re λ0 > ω0(T ), the formula

R(λ0, A) =∫ ∞

0e−λ0sT (s) ds = lim

t→∞

∫ t

0e−λ0sT (s) ds

for the resolvent R(λ0, A) = (λ0I − A)−1 (see e.g. Theorem 34.4 in [45]) shows that

R(λ0, A) ∈ AT . Next, let λ ∈ ρ∞(A) be arbitrary. Taking an appropriate polygon

joining λ0 with λ in ρ∞(A), and considering Taylor series expansions at the vertices,

we get that R(λ,A) ∈ AT also holds.

Choosing an arbitrary z ∈ σ(T ), there exists a unique contractive algebra homo-

morphism hz : AT → C satisfying the condition hz(f(T )) = f(z) (f ∈ Cc(R+)). It

readily follows that hz(R(λ0, A)) = (λ0 − z)−1. Setting w := hz(R(λ,A)), the resolventequation

R(λ0, A)−R(λ,A) = (λ− λ0)R(λ0, A)R(λ,A)

implies that

(λ0 − z)−1 − w = (λ− λ0)(λ0 − z)−1w,

whence w(λ− z) = 1, and it follows that z 6= λ.

We conclude that ρ∞(A) and σ(T ) are disjoint. Thus σ(T ) = C \ ρ∞(A).

Remark 3.2.4. From the proof of the inclusion σ(T ) ⊂ C \ ρ∞(A), one can see that

the above statement also holds if ω0(T ) = −∞ because then σ(A) = ∅.

Corollary 3.2.5. Let T : R+ → L(X) be a C0-semigroup and let A stand for the ge-

nerator of T . Then

σ(T ) ∩ (ω0(T ) + iR) = σ(A) ∩ (ω0(T ) + iR);

that is, the peripheral spectrum of T and that of the generator A are the same.

35

3.3 The regularity constant

In this context, we can reformulate Theorem 2.5.1, which is a generalisation of the

celebrated ArendtBattyLyubichVu theorem.

Theorem 3.2.6. Let T : R+ → L(X) be a C0-semigroup with a p-regular norm-function,

p ∈ Pt. Let A stand for the generator of T . If σ(A) ∩ (ω0(T ) + iR) is countable and

σp(A∗) ∩ (ω0(T ) + iR) is empty then

limn→∞

supt≥0

1n

∫ t+n

t

‖T (s)x‖p(s)

ds = 0

is true for all x ∈ X.


Let us take a C0-semigroup T which is not quasinilpotent; that is, r(T (s)) > 0 for some

(and then for all) s ∈ R+. Then let us introduce the regularity constant κT as it was

done in the discrete case (see [43]):

κT := infn∈N

sups∈R+

(1n

∫ s+n

sr(T (t))−1‖T (t)‖ dt

)(sup

s≤y≤s+nr(T (y))−1‖T (y)‖

)−1

.

Evidently, we have 0 ≤ κT ≤ 1. The regularity constant makes it possible for us to give

a description of semigroups whose norm-function exhibits a regular behaviour.

Our main result is the following.

Theorem 3.3.1. Let T : R+ → L(X) be a C0-semigroup. Then the next conditions are

equivalent:

(i) T has a p-regular norm-function with a topological gauge function p ∈ Pt,

(ii) T has a p-regular norm-function with a continuous gauge p ∈ Pt,

(iii) ‖T (s)‖ ≥ 1 for every s ∈ R+ and κT > 0.

First we will show that it is always possible to nd a continuous gauge function.

Proposition 3.3.2. If T has a p-regular norm-function, p ∈ Pt, then there exists a

continuous gauge q ∈ Pt such that T has regular norm behaviour with respect to q.

36


Proof. Step 1. To get some control over the jumps of the gauge function, in the rst

step we will construct a gauge that is lower semicontinuous.

For every k ∈ Z+, applying Lusin's theorem and the regularity of the Lebesgue

measure, we can choose an open set Ek ⊆ (k, k + 1) such that |Ek| < 2−k and p is a

continuous function on (k, k + 1) \ Ek.By our assumption on p, the constant Gp := supp(s + t)/p(t) : t ∈ R+, s ∈ [0, 1]

is nite; in fact Gp ∈ [1,∞). Let us dene the function v : R+ → [1,∞) in the following

way. For every k ∈ Z+, let

v(s) := p(s) if s ∈ (k, k + 1) \ Ek,

and let

v(s) := inf Gpp(t) : t ∈ [k, s) \ Ek if s ∈ Ek.

Finally v(k) := lim infs→k v(s) for k ∈ Z+.

Note that if s ∈ Ek (k ∈ Z+), then for any t ∈ [k, s)\Ek(6= ∅) we have p(s) ≤ Gpp(t);hence p(s) ≤ v(s) holds for all s ∈ R+ \ Z+. A simple calculation shows also that the

locally bounded function v is lower semicontinuous, and so it is measurable. Since

the norm-function ‖T (s)‖ (s ∈ R+) is also lower semicontinuous we may infer that

‖T (k)‖ ≤ lim infs→k ‖T (s)‖ ≤ lim infs→k v(s) = v(k) for k ∈ Z+. Thus ‖T (s)‖ ≤ v(s)holds for all s ∈ R+. Next, since the set

E := s ∈ R+ : p(s) 6= v(s) ⊆⋃k∈Zk

(Ek ∪ k)

is of nite measure, the function

g(s) :=‖T (s)‖p(s)

− ‖T (s)‖v(s)

(s ∈ R+)

belongs to the class L1(R+) ∩ L∞(R+), and so m(g) = 0 for every m ∈ Mt(R+).Therefore at-lims‖T (s)‖/p(s) 6= 0 implies at-lims‖T (s)‖/v(s) 6= 0.

Given any k ∈ Z+, a brief consideration of the possible cases makes it clear that

v(t + s)/v(t) ≤ Gp holds, whenever k < t ≤ t + s < k + 1. As for the denition of

v in the integer case, we can extend the validity of this inequality to the case when

k ≤ t ≤ t+ s ≤ k + 1. If k ≤ t < k + 1 < t+ s and 0 < s ≤ 1, then

v(s+ t)v(t)

=v(s+ t)v(k + 1)

v(k + 1)v(t)

≤ G2p.

Thus for any t ∈ R+, s ∈ [0, 1] and n ∈ N, we have

v(t+ ns)v(t)

=n∏i=1

v(t+ is)v(t+ (i− 1)s)

≤ G2np .

37


We may conclude that supv(t+ s)/v(t) : t ∈ R+, s ∈ K is nite for any compact set

K ⊆ R+. Furthermore, for every s ∈ R+ the functions |vsv−1−cp(s)| and |psp−1−cp(s)|can dier only on the set E ∪ ((E − s) ∩ R+) of nite measure, and so at-lim|psp−1 −cp(s)| = 0 implies that at-lim|vsv−1 − cp(s)| = 0.

Therefore v is a lower semicontinuous topological gauge function (with cv = cp) and

T has a v-regular norm-function.

Step 2. Next we can construct a piecewise continuous gauge function p adjusted to T .

Let us have an arbitrary η ∈ (0, 1). Since v is lower semicontinuous, for every t ∈ R+,

the set Ut := s ∈ R+ : v(s) > ηv(t) is open in R+; clearly t ∈ Ut. We can introduce

the extended real number τ(t) := sups ∈ (t,∞) : (t, s) ⊆ Ut ∈ (t,∞]. If τ(t) is nite,then v(τ(t)) ≤ ηv(t). As v ≥ 1, it follows that for every t ∈ R+ there exists an n ∈ Nsuch that τn(t) =∞.

Given a k ∈ Z+ there exists a unique nk ∈ N such that τnk−1(k) < k + 1 ≤τnk(k) ∈ [k+ 1,∞]. Next, let us introduce the notation tk,i := τ i(k) for 0 ≤ i < nk, and

tk,nk := k + 1. For every 0 ≤ i < nk, we have

ηv(tk,i) ≤ v(t) ≤ Gvv(tk,i) for all t ∈ [tk,i, tk,i+1).

By Lusin's theorem, there exists a continuous function

gk,i : [tk,i, tk,i+1)→ [ηv(tk,i), Gvv(tk,i)],

which diers from the restriction of v to this interval in a set Ek,i of measure less than

(2nk)−12−k. Selecting tk,i < tk,i < tk,i < tk,i+1 such that max(tk,i − tk,i, tk,i+1 − tk,i) <(4nk)−12−k, we set hk,i(t) := gk,i(tk,i) if tk,i ≤ t ≤ tk,i, hk,i(t) := gk,i(t) if tk,i ≤ t ≤ tk,iand hk,i(t) := gk,i(tk,i) if tk,i ≤ t < tk,i+1.

Note that the continuous function hk,i diers from the restriction of v in a set Hk,i

of measure less than n−1k 2−k; furthermore the range of hk,i is contained in the range of

gk,i.

Let us dene the piecewise continuous function h on R+ so that h coincides with

hk,i on the interval [tk,i, tk,i+1) for all k ∈ Z+ and 0 ≤ i < nk. It is clear that h diers

from v on a set of nite measure. Furthermore, if t, s ∈ [k, k + 1) and t < s (k ∈ Z+),then t ∈ [tk,i, tk,i+1) and s ∈ [tk,j , tk,j+1) for some 0 ≤ i ≤ j < nk, and so

h(s)h(t)

≤Gvv(tk,j)ηv(tk,i)

≤Gvη

j−iv(tk,i)ηv(tk,i)

≤ Gvη.

If t ∈ [tk,i, tk,i+1) ⊆ [k, k + 1) and s = k + 1, then

h(s)h(t)

≤ Gvv(k + 1)ηv(tk,i)

≤ G2v

η.

38

3.4 One consequence of regularity

We nd as in Step 1 that suph(t + s)/h(t) : t ∈ R+, s ∈ K is nite for any compact

subset K of R+.

The piecewise continuous function w := η−1Gvh preserves the latter property. Fur-

thermore, for any t ∈ [tk,i, tk,i+1) (k ∈ Z+, 0 ≤ i < nk), we have that

w(t) = η−1Gvh(t) ≥ η−1Gvηv(tk,i) ≥ η−1GvηG−1v v(t) = v(t);

that is w ≥ v. Taking into account the fact that w diers from η−1Gvv on a set of nite

measure, we may deduce as in Step 1 that w ∈ Pt and T has a w-regular norm-function.

Step 3. We complete the proof by associating a continuous topological gauge function

q with T.

The function w has countable discontinuity points, and we may assume that it has

innitely many. We place these points into the strictly increasing sequence tnn∈N;clearly 0 < t1 and limn→∞ tn =∞. For every n ∈ N, let δn denote the minimum of the

positive numbers 3−1(tj − tj−1)n+1j=1 (here t0 := 0). We know that w is continuous to

the right of tn and that is has a left limit w(tn − 0) := limt→tn−0w(t). Let N+ stand

for the set of positive integers n, where w(tn − 0) < w(tn), and N− := N \N+. For any

n ∈ N+, we can nd a real number rn such that 0 < tn − rn < min(δn, 2−n) and w is

constant on the interval [rn, tn). The function q is dened on [rn, tn] so that it is linear

and q(rn) := w(rn), q(tn) := w(tn). If n ∈ N−, then we can choose an rn such that

0 < rn − tn < min(δn, 2−n) and w is constant on the interval [tn, rn]. The function q islinear on [tn, rn] and satises the conditions q(tn) = w(tn−0) and q(rn) = w(rn). Next,for every t ∈ R+ \ ((

⋃n∈N+

[rn, tn]) ∪ (⋃n∈N−

[tn, rn])), we will set q(t) := w(t).It immediately follows that q is a continuous function, q ≥ w, and q diers from w

on a set of positive measure. Now let t, s ∈ R+ be arbitrary and satisfy 0 < s− t ≤ 1.If t ∈ (rj , tj), s ∈ (rk, tk) (j, k ∈ N+, j ≤ k), then

q(s)q(t)

≤ q(tk)q(rj)

=w(tk)w(rj)

≤ G2w

since tk − rj ≤ 2. The reader can readily check that q(s)/q(t) ≤ G2w holds in other

possible cases too. We get as before that q ∈ Pt and T has a q-regular norm-function.


Now we will prove that the regularity of the norm-function implies the positivity of the

regularity constant.

39


Proposition 3.4.1. Let T : R+ → L(X) be a C0-semigroup with a p-regular norm

function, where p ∈ Pt is continuous. Then ‖T (s)‖ ≥ 1 for every s ∈ R+ and the

regularity constant κT is positive.

Proof. Let T be a C0-semigroup with p-regular norm function, where p ∈ Pt is continu-

ous. Changing T (s) to r(T (s))−1T (s), we may assume that cp(s) = cT (s) = r(T (s)) = 1(s ∈ R+). If ‖T (s0)‖ < 1 for some s0 ∈ (0,∞), then lims ‖T (s)‖ = 0, which contradicts

the assumption at-lims‖T (s)‖/p(s) 6= 0. Therefore ‖T (s)‖ ≥ 1 holds for all s ∈ R+.

Supposing that κT = 0 we need to show that it leads to a contradiction. For any

s ∈ R+ and n ∈ N let us introduce the notation

A(s, n) :=1n

∫ s+n

s‖T (t)‖ dt, B(s, n) := sup‖T (t)‖ : t ∈ [s, s+ n]

and

C(s, n) :=1n

∫ s+n

s

‖T (t)‖p(t)

dt, D(s, n) :=1n

∫ s+n

s

∣∣∣∣p(t+ 1)p(t)

− 1∣∣∣∣ dt.

We infer by Proposition 3.1.7 that α := lim supn supC(s, n) : s ∈ R+ ∈ (0, 1]; let usgive an arbitrary α ∈ (0, α). Let us choose a positive ε which satises the condition

4G2pε <

α

3,(3.2)

where Gp := supp(t + s)/p(t) : t ∈ R+, s ∈ [0, 1]. Since κT = 0, it follows that thereexists k ∈ N such that

A(s, k)B(s, k)

≤ ε for all s ∈ R+.(3.3)

Let us give a positive η and a positive integer N satisfying the conditions√2η < min

(α

3k,k+1√

2− 1, 1− 1k+1√

2

)(3.4)

and

2N−1 <α

3.(3.5)

Since α < α and limn supD(s, n) : s ∈ R+ = 0 by Proposition 3.1.7, we can nd l ∈ Nand s ∈ R+ such that

l/k > N, C(s, l) > α and D(s, l) < η.(3.6)

40


The integer l can be written in the form l = nk + r, where n := [l/k] ≥ N and r ∈ Z+,

r < k. Applying (3.5) and (3.6), we have

α <k

l

n−1∑j=0

C(s+ jk, k) +r

lC(s+ nk, r)

≤ 2kl

+k

l

n−1∑j=1

C(s+ jk, k)

<α

3+k

l

n−1∑j=1

C(s+ jk, k).

(3.7)

From (3.6) it also follows that

n−1∑j=1

D(s+ jk − 1, k + 1) ≤ 2lD(s, l)(k + 1)−1 < 2η(k + 1)−1l.(3.8)

Next, let N+ (respectively, N−) stand for the set of those positive integers j which

satisfy the conditions j < n and D(s + jk − 1, k + 1) ≥√

2η(k + 1)−1 (respectively,

j < n and D(s+ jk − 1, k + 1) <√

2η(k + 1)−1). We get from (3.8) that

(card N+)√

2η(k + 1)−1 ≤∑j∈N+

D(s+ jk − 1, k + 1) < 2η(k + 1)−1l

whence

card N+ < l√

2η(3.9)

follows.

Fixing an index j ∈ N−, we dene ω+ (respectively, ω−) as the set of points t in the

interval [s+ jk− 1, s+ (j + 1)k] which satisfy the condition |p(t+ 1)p(t)−1 − 1| ≥√

2η(respectively, |p(t+ 1)p(t)−1 − 1| <

√2η). The inequality√

2η|ω+| ≤ (k + 1)D(s+ jk − 1, k + 1) <√

2η

implies |ω+| < 1; thus there exists s0 ∈ (s+ jk − 1, s+ jk) such that s0 + i ∈ ω− holds

for every non-negative integer i with i ≤ k. For any t ∈ ω− we have by (3.4) that

1k+1√

2< 1−

√2η <

p(t+ 1)p(t)

< 1 +√

2η < k+1√

2.

Hence for every integer i with 1 ≤ i ≤ k + 1 we get

2−1 ≤ 2−i

k+1 ≤ p(s0 + i)p(s0)

=i−1∏v=0

p(s0 + v + 1)p(s0 + v)

≤ 2i

k+1 ≤ 2.

41

3.5 The converse statement

Thus

2−1p(s0) ≤ p(s0 + i) ≤ 2p(s0) whenever 0 ≤ i ≤ k + 1.(3.10)

Given any t1, t2 ∈ [s+jk, s+(j+1)k] there exist integers i1, i2 ∈ [0, k] and real numbers

a1, a2 ∈ [0, 1) such that t1 = s0 + i1 + a1 and t2 = s0 + i2 + a2. Applying (3.10) we nd

that

p(t1)p(t2)

=p(s0 + i1 + a1)p(s0 + i1)

p(s0 + i1)p(s0 + i2 + 1)

p(s0 + i2 + 1)p(s0 + i2 + a2)

≤ 4G2p.

(3.11)

Assuming that p takes its maximum on the interval [s+ jk, s+ (j + 1)k] at t∗, we inferby (3.11), (3.3) and (3.2) that

C(s+ jk, k) ≤ 1k

∫ s+(j+1)k

s+jk

‖T (t)‖B(s+ jk, k)

p(t∗)p(t)

dt

≤ 4G2p

A(s+ jk, k)B(s+ jk, k)

≤ 4G2pε <

α

3.

(3.12)

Finally, by the use of (3.7), (3.12), (3.9) and (3.4) we may conclude that

α <α

3+k

l

∑j∈N+

C(s+ jk, k) +k

l

∑j∈N−

C(s+ jk, k)

≤ α

3+k

l(card N+) +

k

l(card N−)

α

3

≤ α

3+k

ll√

2η +k

lnα

3<

α

3+α

3+α

3= α,

which is a contradiction.


We will begin with an observation which will be useful in the proof of the suciency of

our regularity condition.

Lemma 3.5.1. Let T : R+ → L(X) be a non-quasinilpotent C0-semigroup such that

κT > 0. Then for every n ∈ N and x ∈ R+ there exists an x0 > x such that

1n

∫ x0+n

x0

r(T (s))−1‖T (s)‖supy∈[x0,x0+n] ‖r(T (y))−1T (y)‖

ds ≥ κT2.

42


Proof. We shall use an indirect argument: suppose that the statement is not true for

some n and x. By the denition of κT , we can choose m ∈ N and sm ∈ R+ such that

m−1(xn−1 + 1) < κT /4 and

I :=1mn

∫ sm+mn

sm

‖r(T (s))−1T (s)‖supy∈[sm,sm+mn] ‖r(T (y))−1T (y)‖

ds >3κT

4.

Let l stand for the integer part of max(0, (x− sm)/n). If k > l then sm + kn > x. Thus

I ≤ 1m

m−1∑k=0

1n

∫ sm+(k+1)n

sm+kn

‖r(T (s))−1T (s)‖supy∈[sm+kn,sm+(k+1)n] ‖r(T (y))−1T (y)‖

ds

≤ l + 1m

+m− 1− l

m

κT2

≤ xn−1 + 1m

+κT2

<κT4

+κT2

=3κT

4.

This is a contradiction, so the lemma must be true.

We are now in a position to prove the remaining implication.

Proposition 3.5.2. Let T : R+ → L(X) be a C0-semigroup such that ‖T (s)‖ ≥ 1 for

every s ∈ R+ and κT > 0. Then T has a p-regular norm-function with some p ∈ Pt.

Proof. Rescaling the semigroup, we may suppose again that eω0(T )s = r(T (s)) = 1(s ∈ R+). This means that for every ω ∈ (0,∞) there exists an Mω ∈ [1,∞) such that

‖T (s)‖ ≤ Mωeωs for all s ∈ R+. Then for every n ∈ N we can choose a kn ∈ N such

that kn > n and ‖T (t)‖ ≤ et/n whenever t ∈ [kn,∞). Note that for any s ∈ R+

(3.13) ‖T (t)‖ ≤ ‖T (s)‖‖T (t− s)‖ ≤ ‖T (s)‖e(t−s)/n whenever t ∈ [s+ kn,∞).

Next, dene an l ∈ N with the condition

l > 4/κT .(3.14)

We shall again use the notation A(s, n), B(s, n) introduced in the proof of Proposi-

tion 3.4.1. From the denition of κT we can choose an s1 ∈ R+ such that A(s1, (l +1)k1)/B(s1, (l+1)k1) ≥ κT /2. Assuming that sini=1 (n ∈ N) has been already dened,

by Lemma 3.5.1 we can nd an sn+1 ∈ R+ which satises the conditions

sn+1 > sn + n(l + 1)(kn + kn+1) + n(n+ 1)(3.15)

43


and A(sn+1, (l + 1)kn+1)/B(sn+1, (l + 1)kn+1) ≥ κT /2. By induction, we obtain a se-

quence sn∞n=1 such that sn+1 > sn + (l + 1)kn and

A(sn, (l + 1)kn)/B(sn, (l + 1)kn) ≥ κT /2 for all n ∈ N.(3.16)

Now the function p : R+ → [1,∞) is dened in the following way. If s1 > 0 then

p(t) := sup‖T (s)‖ : s ∈ [0, s1) for every t ∈ [0, s1). Furthermore, for every n ∈ N we

will set

p(t) :=

B(sn, (l + 1)kn) if t ∈ [sn, sn + lkn],

B(sn, (l + 1)kn)e(t−sn−lkn)/n if t ∈ [sn + lkn, sn+1).(3.17)

The locally bounded, measurable function p clearly dominates the norm function

of T on the intervals [0, s1) and [sn, sn + (l + 1)kn] (n ∈ N). On the other hand, if

t ∈ (sn + (l + 1)kn, sn+1) (n ∈ N) then by (3.13) and (3.17) we have

‖T (t)‖ ≤ ‖T (sn + lkn)‖e(t−sn−lkn)/n ≤ p(t).

Thus ‖T (t)‖ ≤ p(t) holds for all t ∈ R+. Applying the relations

A(sn, lkn) =l + 1lA(sn, (l + 1)kn)− 1

lA(sn + lkn, kn)

and B(sn, (l + 1)kn) ≥ B(sn + lkn, kn), by (3.17), (3.16) and (3.14) we nd that

1lkn

∫ sn+lkn

sn

‖T (t)‖p(t)

dt =A(sn, lkn)

B(sn, (l + 1)kn)

≥ A(sn, (l + 1)kn)B(sn, (l + 1)kn)

− 1l

A(sn + lkn, kn)B(sn + lkn, kn)

≥ κT2− 1l>κT4

is true for every n ∈ N. Since [0, lkn]n is a Følner sequence, we may infer from

Proposition 3.1.7 that at-lims‖T (s)‖/p(s) = 0 does not hold.

Now let t, s ∈ R+ be arbitrary. If sn ≤ t ≤ t+ s < sn+1 holds for some n ∈ N, thenit immediately follows that 1 ≤ p(t+ s)/p(s) ≤ es/n, hence∣∣∣∣p(t+ s)

p(s)− 1∣∣∣∣ ≤ es/n − 1.(3.18)

Let us next assume that

sn+1 − n ≤ t < sn+1 ≤ t+ s ≤ sn+1 + n

44

3.6 An example for a gauge function

is valid for some n ∈ N. Since sn+1−n > sn + (l+ 1)kn by (3.15), it follows from (3.17)

that

p(t) = Qn exp(t− sn − lkn

n

)≥ Qn exp

(sn+1 − n− sn − lkn

n

),(3.19)

where Qn := B(sn, (l + 1)kn). On the other hand, sn+1 > sn + lkn + kn+1 is also true

by (3.15), and so we infer by (3.13) that

‖T (u)‖ ≤ ‖T (sn + lkn)‖ exp(u− sn − lkn

n+ 1

)≤ Qn exp

(sn+1 + (l + 1)kn+1 − sn − lkn

n+ 1

)holds for every u ∈ [sn+1, sn+1 + (l + 1)kn+1]. Thus

p(t+ s) = Qn+1 ≤ Qn exp(sn+1 + (l + 1)kn+1 − sn − lkn

n+ 1

).(3.20)

Applying (3.15), (3.19) and (3.20), a short computation shows that p(t + s) ≤ p(t),hence ∣∣∣∣p(t+ s)

p(s)− 1∣∣∣∣ ≤ 1.(3.21)

Taking into account the fact that sn+1 − sn > n2 tends to innity, the relations (3.18)

and (3.21) imply that supp(t + s)/p(t) : t ∈ R+, s ∈ K is nite for every compact

subset K of R+. Furthermore, in view of Proposition 3.1.7 we may conclude that

at-limt

∣∣∣∣p(t+ s)p(t)

− 1∣∣∣∣ = 0 for every s ∈ R+.

Therefore p is a topological gauge function and the C0-semigroup T has regular norm

behaviour with respect to p.


Now we will show that condition (iii) is not a consequence of other properties in the

denition of a topological gauge function. We will construct a locally bounded, mea-

surable function p : R+ 7→ [1,∞) such that for every s ∈ R+ ps/p is bounded and

at-limt|p(s+ t)/p(t)− 1| = 0, but sups∈[0,1] supt∈R+p(s+ t)/p(t) is innite. Next, let us

dene a p in the following way:

45


p(t) :=

1 + 1

n(t− 2n) if 2n ≤ t < 2n + n2,

1 if 2n + n2 ≤ t < 2n + n2 + 1n ,

1 + n2(t− 2n − n2 − 1n) if 2n + n2 + 1

n ≤ t ≤ 2n + n2 + 2n ,

1 if 2n + n2 + 2n < t ≤ 2n+1.

Evidently, the mapping t 7→ p(t + s)/p(t) is bounded for every s ∈ R+. Moreover,

limt→∞ p(t + s)/p(t) = 1 for every s ∈ [0, 1] except for the set ∪∞n=1[2n + n2 − 1, 2n +

n2 + 1] which has a zero upper Banach density. Now it is easy to verify that at-

limt|p(s+ t)/p(t)− 1| = 0, so cp(s) = 1 (s ∈ R+). On the other hand,

p 1n

(2n + n2 + 1n)

p(2n + n2 + 1n)

= n;

that is, sups∈[0,1] supt∈R+p(s+ t)/p(t) =∞.

This example suggests the following question.

Question 3.6.1. Let T : R+ → L(X) be a C0-semigroup. Let p : R+ → [1,∞) be

a measurable, locally bounded function such that ps/p is bounded and topologically

almost converges in the strong sense to a positive real number cp(s) for every s ∈ R+.

Next, assume that ‖T (s)‖ ≤ p(s) holds for every s ∈ R+, and that at-lims‖T (s)‖/p(s) =0 does not hold. Does it then follow that T has a topologically regular norm-function?

46

Chapter 4

A KatznelsonTzafriri type result

4.1 Introduction

Let us consider a complex Banach space X and let T be a power-bounded operator on X,

i.e. supn≥1 ‖Tn‖ <∞ holds. Then it is a simple matter to give a characterization of the

norm stability of T (that is limn→∞ ‖Tn‖ = 0) via the spectral radius formula. Indeed,

r(T ) < 1 if and only if ‖Tn‖ (n→∞) tends to zero, or equivalently ‖Tn‖ tends to zeroexponentially; that is, there exists a C > 0 and λ ∈ (0, 1) such that ‖Tn‖ ≤ Cλn for

every n ∈ N. However, it is more interesting to give necessary and sucient conditions

for the uniform convergence

limn→∞

‖TnQ‖ = 0

with some bounded operator Q, as this is far from trivial. The case Q = T − I,

which is of great importance because of its role in the uniform zerotwo law (cf. [34])

and its connection with the GelfandHille theorem (see [68]), was rst characterized

by J. Esterle [22]. Moreover, if the sequence ‖Tn − Tn+1‖ tends to zero as n → ∞,the convergence may be far from exponential. On the quantitative behaviour of this

sequence we refer the reader to [55, Chapter 4] and [22].

We recall that the celebrated KatznelsonTzafriri theorem asserts that if T is a

power-bounded operator acting on a Banach space X, and f(z) =∑∞

k=0 akzk is a power

series with absolute convergent coecients, which is of spectral synthesis with respect to

the peripheral spectrum of T, then limn→∞ ‖Tnf(T )‖ = 0 (see [34]). In a Hilbert space

setting a richer functional calculus can be dened for contractions due to von Neumann's

inequality. We recall that the disk algebra A(D) is the set of all holomorphic functions

47

4.2 Preliminaries and the main result

in the unit disc which extend continuously to the unit circle. Actually, the following

was proved in [23].

Theorem 4.1.1. Let T be a contraction in a Hilbert space and let f ∈ A(D). Thenf(λ) = 0 for every λ ∈ σ(T ) ∩ T if and only if limn→∞ ‖Tnf(T )‖ = 0.

On the other hand, contractions even admit a more general H∞ calculus on the unit

disc. This fact was exploited by H. Bercovici, who proved the following ([9]).

Theorem 4.1.2. Let T be a completely nonunitary contraction on a Hilbert space, and

let f be a bounded holomorphic function on the unit disc. Then limn→∞ ‖Tnf(T )‖ = 0holds if limr→1 f(reiθ) = 0 for every eiθ ∈ σ(T ) ∩ T.

An example shows that the converse implication of the theorem is not true (see

[9]). The statement of Theorem 4.1.1 was recently extended by S. Mustafayev, who

considered the Banach algebra AT generated by a contraction T. In fact, he proved in

[52, Theorem 3.7] that if T is a contraction on a Hilbert space such that its peripheral

spectrum σ(T )∩T has zero Lebesgue measure, then the Gelfand transform of R ∈ A(T )

vanishes on σ(T ) ∩ T if and only if limn→∞ ‖TnR‖ = 0.

Here we shall prove that the assumption of the KatznelsonTzafriri theorem can

be weakened in Hilbert spaces, and we shall present a characterization of the condition

limn→∞ ‖TnQ‖ = 0 if Q commutes with T, via an ergodic condition. We shall also prove

that, if f ∈ A+(T) and Q = f(T ), the given condition is equivalent to the vanishing of

f on the peripheral spectrum of T.

The proof shall partly follow Vu's method ([64], [65]); that is, we will rst verify

convergence in the strong operator topology by reducing the problem to isometries.

After that we shall complete the proof using some aspects of an ultrapower approach.


Let L(X) be the algebra of all bounded linear operators on a Banach space X. Let σ(T )

stand for the spectrum of T ∈ L(X), and let I be the identity operator on X. Next, let

A+(T) denote the set of sums of power series on the unit circle T whose coecients are

absolutely convergent. Then, for each power-bounded operator T , a bounded functional

calculus naturally arises which is dened by f(T ) :=∑∞

k=0 akTk ∈ L(X), where f(z) =∑∞

k=0 akzk and

∑∞k=0 |ak| <∞.

Our main result will now be presented.

48


Theorem 4.2.1. Let T be a power-bounded operator on a Hilbert space H. If Q ∈ L(H)and TQ = QT, then the following statements are equivalent:

(i) limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT kQ

∥∥∥∥∥ = 0 for every λ ∈ σ(T ) ∩ T,

(ii) limn→∞ ‖TnQ‖ = 0.

Moreover, if Q = f(T ) for some f ∈ A+(T), then (i) and (ii) are equivalent to

(iii) f(λ) = 0 for every λ ∈ σ(T ) ∩ T.

First we will prove a lemma which leads us to the ergodic condition of the above

theorem.

Lemma 4.2.2. Let T be a power-bounded operator on a complex Banach space X and

let f ∈ A+(T). Then, for every λ ∈ T, we have

limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT k (f(T )− f(λ)I)

∥∥∥∥∥ = 0.

Proof. Using the Taylor expansion f(z) =∑∞

m=0 amzm of f, and changing the order of

summation, we have

1n

n−1∑k=0

λ−kT k (f(T )− f(λ)I) =1n

n−1∑k=0

λ−kT k

( ∞∑m=0

am(Tm − λmI)

)

=∞∑m=0

amλm

(1n

n−1∑k=0

(λ−(k+m)T k+m − λ−kT k)

).

Next, pick an ε > 0 and let us choose an indexM ∈ N such that∑∞

m=M |am| ≤ ε. Thenwe may infer that (with 4 denoting the symmetric dierence of two sets)

∥∥∥∥∥ 1n

n−1∑k=0


∥∥∥∥∥ ≤M−1∑m=0

|am|

∥∥∥∥∥ 1n

n−1∑k=0


∥∥∥∥∥+

∞∑m=M

|am|

∥∥∥∥∥ 1n

n−1∑k=0


∥∥∥∥∥≤ ‖f‖1L max

1≤m≤M

card(1, ..., n4m+ 1, ...,m+ n)n

+ 2Lε

≤ ‖f‖12LMn−1 + 2Lε < 4Lε

49


provided n is suciently large, where ‖f‖1 :=∑∞

m=0 |am| and L := sup‖T k‖ : k =0, 1, 2, . . .. Since ε was arbitrarily chosen, the lemma is proved.

The uniform ergodic theorem tells us that 1n

∑n−1k=0 T

k tends to zero in norm if and

only if 1 is in the resolvent set of T (cf. [44, Theorem 2.7]). With this result, the

following corollary of the above lemma is straightforward.

Corollary 4.2.3. Let T be a power-bounded operator on a Banach space X and f ∈A+(T). Then, for each λ ∈ σ(T ) ∩ T,

f(λ) = 0 if and only if limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT kf(T )

∥∥∥∥∥ = 0.

In the following lemma we shall prove the theorem for an arbitrary isometry. Recall

that Douglas's extension theorem ([16]) states that any isometry V on a Banach space

X can be extended to a surjective isometry on a larger space. This result was already

well known in the Hilbert space setting (cf [32]). Later, C.J.K. Batty and S. Yeates

gave a dierent construction in [7] which preserves the structure of the original space

in most cases (for instance, when X is a Hilbert space or superreexive). In addition,

their construction makes it possible for one to dene an extension of the commutant

of V ; they showed that there exists a unital isometric algebra homomorphism from

the commutant of V into the commutant of its surjective extension [7, Proposition

3.5]. Then, under this homomorphism, the spectrum of an element of the commutant

contains the spectrum of its image. (The reader should see Bercovici's paper [10] for

a similar construction in the Hilbert space case.) Below, we shall make use of these

results.

Lemma 4.2.4. Let V be an isometry on a Hilbert space H. Suppose that Q ∈ L(H)and QV = V Q. If

limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kV kQ

∥∥∥∥∥ = 0

holds for every λ ∈ σ(V ) ∩ T, then Q = 0.

Proof. Applying the extension theorem of Batty and Yeates, throughout the proof we

can assume, without loss of generality, that V is unitary.

It can easily be seen that V Q∗ = Q∗V, hence the C∗-algebra A generated by V

and QQ∗ is commutative. Now let Σ stand for the Gelfand spectrum of A and let

50


ΓA : A → C(Σ), A 7→ A be the Gelfand transform of A. Then the ergodic condition

limn→∞1n

∥∥∥∑n−1k=0 λ

−kV kQQ∗∥∥∥∞

= 0 is satised for every λ in the range of V because

ran V = σ(V ).Next, let us assume that the non-negative function QQ∗ ∈ C(Σ) takes its maximum

q0 at h0 ∈ Σ. Choosing λ0 := V (h0), we get for any n ∈ N that

1n

∥∥∥∥∥n−1∑k=0

λ−k0 V kQQ∗

∥∥∥∥∥∞

≥ 1n

∣∣∣∣∣n−1∑k=0

λ−k0 V (h0)k QQ∗(h0)

∣∣∣∣∣ = q0.

This means that q0 = 0 must hold, so QQ∗(h) = 0 for every h ∈ Σ. Hence σ(QQ∗) = 0and QQ∗ = 0. This readily implies that Q = 0.

Let J be an innite set, and U let be a free ultralter on J . Now consider the Banach

spaces `∞(J,X) of all bounded functions from J to X, and c0(J,X; U) of all bounded

functions from J to X which converge to zero through the ultralter. The quotient

space `∞(J,X)/c0(J,X; U) is an ultrapower of the Banach space X, which we shall

denote by XU. Recall that if X is Hilbert space then XU is also a Hilbert space. In fact,

the parallelogram identity is preserved by taking the ultrapower and the polarization

identity tells us that the inner product on XU is given by

〈x, y〉U = U- limn〈xn, yn〉,

where x and y denote the equivalence classes of xnn and ynn in XU, respectively.

For each T ∈ L(X), the ultrapower TU of T is dened by the formula

TU(xn+ c0(J,X; U)) := Txn+ c0(J,X; U).

Note that the mapping T 7→ TU is an isometric unital algebra homomorphism from

L(X) into L(XU) such that σ(T ) = σ(TU) (cf. [27], [60, Theorem V.1.4]).

Proof of Theorem 4.2.1. The implication (ii) =⇒ (i) is evident. To prove (i) =⇒ (ii),

let us introduce a new semi-inner product on H by

〈x, y〉T := m(〈Tnx, Tny〉n) (x, y ∈ H),

where m denotes a Banach limit. Forming quotient space and completion result a

Hilbert space HT , where T acts as an isometry V. If X denotes the canonical embedding

of H into HT , we get that V X = XT. In addition, the intertwining transformation X

naturally induces a contractive unital algebra homomorphism between the commutants

51


of T and V. That is, if A is in T′, the commutant of T, then there exists a unique

operator B ∈ V ′ such that XA = BX and ‖B‖ ≤ ‖A‖. It can be also shown that the

mapping

γT : T′ → V ′, A 7→ B

is a contractive unital algebra homomorphism and hence σ(B) ⊆ σ(A) ([35]). Then (i)

implies that V and γT (Q) satisfy the ergodic condition

limn→∞

n−1

∥∥∥∥∥n−1∑k=0

λ−kV kγT (Q)

∥∥∥∥∥ = 0

for every λ ∈ σ(V ) ∩ T. We can apply now Lemma 2.4 to this and then conclude that

γT (Q) = 0. Thus m(‖QTnx‖2n) = m(‖TnQx‖2n) = 0 for every x ∈ H, and hence

infn≥0 ‖TnQx‖ = 0 by the positivity of m. Therefore, condition (i) in Theorem 4.2.1

implies that infn≥0 ‖TnQx‖ = limn→∞ ‖TnQx‖ = 0 for every x ∈ H.

Now let α := infn≥0 ‖TnQ‖ and L := supn≥0 ‖Tn‖. Choose a sequence of unit

vectors xk∞k=0 in H such that ‖T kQxk‖ ≥ α/2 holds for every k ∈ N. Then, of course,‖TnQxk‖ ≥ α/(2L) for every n ≤ k. Taking a free ultralter U on N, we can dene a

unit vector x = xkk + c0(N,H; U) in the ultrapower space HU. Then, for any n ∈ N,we have

‖TnUQUx‖ = ‖TnQxkk + c0(N,H; U)‖ = U- limk‖TnQxk‖

≥ lim infk‖TnQxk‖ ≥

α

2L.

However, applying the previous conclusion to TU and QU which satisfy (i), we nd that

TnUQU tends to zero strongly. This means that αmust be zero, hence limn→∞ ‖TnQ‖ = 0also holds which is we intended to show.

For (i) ⇐⇒ (iii), see Corollary 4.2.3.

We should mention here that condition (i) implies the convergence

limn→∞

‖TnQ‖ = 0

but limn→∞ ‖Tn|(ranQ)−‖ = 0 does not hold in general (here T|(ranQ)− denotes the res-

triction of T to (ran Q)−, the closure of the range of Q); see, for instance, [33, Example

1.8]. Actually, the structure of the orbits can be quite complicated on (ranQ)−, they may

decay arbitrarily slowly due to a result by V. Müller [53]. More precisely, if σ(T|(ranQ)−)∩T 6= ∅ (i.e. ‖Tn|(ranQ)−‖ 6→ 0) then for every positive sequence an∞n=1 ∈ c0(N) we can

nd a vector h ∈ (ranQ)− such that ‖Tnh‖ ≥ an for every n ∈ N.

52


Remark 4.2.5. We note that the local version of Theorem 4.2.1 does not hold. In fact,

let us consider the Hilbert space `2(N), and let T be the right forward shift on `2(N)and let Q be the identity operator. Choosing the unit vector e0 := (1, 0, 0, . . .), it canbe easily seen that

limn→

1n

∥∥∥∥∥n−1∑k=0

λkT ke0

∥∥∥∥∥ = 0

for every λ ∈ T, but 1 = ‖Tne0‖ 6→ 0 if n→∞.

Remark 4.2.6. Note that Lemma 4.2.4 appears to be crucial in the proof of the above

theorem. We do not know whether a similar statement could be veried in more general

spaces. The extension theorem given by Batty and Yeates preserves the structure of

superreexive spaces, hence it seems to be natural to ask whether Lemma 4.2.4 can be

generalized to superreexive spaces. It is an open problem whether the ergodic result

Theorem 4.2.1 is true in Lp-spaces (1 < p <∞), or in superreexive spaces.

53

Chapter 5

Summary

In essence, this dissertation is a study on the stability properties of strongly continuous

operator semigroups. I provide generalizations of well-known stability results including

the ArendtBattyLyubichVu theorem and the KatznelsonTzafriri Theorem. Fur-

thermore, I give a characterization of C0-semigroups whose norm function is topologi-

cally regular. These results are based on three recent papers [42], [46] and [47].

5.1 Representations of locally compact commutative semi-

groups

This part of the thesis is based on a joint paper with László Kérchy.

Consider a locally compact, Hausdor abelian group (G; +). Let S be a closed

subsemigroup of G with non-empty interior S such that S − S = G and S ∩ (−S) =

0. Let S] stand for the set of characters of S. In Chapter 2, by using invariant

means on S, we introduced the concept of gauge functions, representations with regular

norm-function, and we also studied the properties of limit functionals. We proved the

following.

Theorem. Let p be a gauge function on S and let us assume that there exists a repre-

sentation ρ : S → L(X) with a p-regular norm-function. Then the limit functional cp of

p is a positive character of S.

Theorem. If the representation ρ : S → L(X) has regular norm-behaviour with respect

to the gauge functions p and q, then

cp = cq.

54

5.1 Representations of locally compact commutative semigroups

The above theorem leads to the following denition. The function cρ := cp is called

the limit functional of the representation ρ with p-regular norm-function.

When S = R+, we proved that if T : R+ → L(X) is a representation that has a

regular norm-function then the spectral radius function of T is equal to cT .

We recall that Cc(S) stands for the set of continuous functions with compact support

in S. The Fourier transform of a function f ∈ Cc(S) with respect to the representation

ρ : S → L(X) is given by

f(ρ) :=∫Sf(s)ρ(s) dµ(s).

We can similarly dene f(χ) when χ ∈ S], since the characters of S are one-dimensional

representations. In Chapter 2 we introduced dierent kinds of spectra of unbounded

representations and studied connections among them, which provide an interplay with

already known concepts.

The peripheral spectrum of the representation ρ : S → L(X) with regular norm-

function is dened by

σper(ρ) := χ ∈ σ(ρ) : |χ| = cρ and |χ(s)| = cρ(s) for all s ∈ S .

Let the point spectrum of the representation ρ : S → L(X) be the set

σp(ρ) :=χ ∈ S] : there exists 0 6= x ∈ X with ρ(s)x = χ(s)x for all s ∈ S

.

The following statement is one of the main results of this dissertation, which is an

ArendtBattyLyubichVu type theorem for representations having a regular norm-

function. Instead of the usual stability of orbits, we shall apply a slightly weaker concept

of stability that may be described by using almost convergence.

Theorem. Let ρ : S → L(X) be a representation with a p-regular norm-function. If

σper(ρ) is countable and σp(ρ∗)∩χ ∈ S] : |χ| = cρ is empty, then ‖ρ(s)x‖/p(s) almost

converges to zero for all x ∈ X.

A corollary of the theorem is the following.

Corollary. Let ρ : S → L(X) be a representation with a p-regular norm-function. If

σper(ρ) is countable and σp(ρ∗) ∩ χ ∈ S] : |χ| = cρ is empty, then

limi→∞

1µ(Ki)

∫Ki

‖ρ(s)x‖p(s)

dµ(s) = 0

is true for all x ∈ X, where Kii is any Følner sequence.

55



In Chapter 3 we focused on the representations of the real half line. Here, similar to the

previous chapter, we introduced C0-semigroups with a topological norm-function. This

can be achieved using the set of topologically invariant means instead of the translation

invariant means.

The main result of this chapter provides a characterization of C0-semigroups whose

norm-function is topologically regular. The statement is similar to the characterization

proved by Kérchy and Müller [38], [43] in the discrete case. Let us assume that (T (s))s≥0

is a C0-semigroup and the spectral radius of T (s) is positive, i.e. r(T (s)) > 0, for every

s ∈ R+. Let us dene

κT := infn∈N

sups∈R+

( 1n

∫ s+n

sr(T (t))−1‖T (t)‖ dt

)(sup

s≤y≤s+nr(T (y))−1‖T (y)‖

)−1 .

Let Pt denote the set of topological gauge functions with respect to T. The following

theorem is the main result here.

Theorem. Let T : R+ → L(X) be a C0-semigroup. Then the following conditions are

equivalent:

(i) T has a p-regular norm-function with a topological gauge function p ∈ Pt,

(ii) T has a p-regular norm-function with a continuous gauge p ∈ Pt,

(iii) ‖T (s)‖ ≥ 1 for every s ∈ R+ and κT > 0.

5.3 A KatznelsonTzafriri type theorem in Hilbert spaces

The KatznelsonTzafriri theorem is an operator-theoretic result which is related to the

ABLV theorem. When S = Z+, we can present an extension of the result in the

Hilbert space setting. Our aim is to characterize, via an ergodic condition, the norm

convergence limn→∞ ‖TnQ‖ = 0 when T is a power-bounded operator on a Hilbert

space and Q commutes with T.

Let I be the identity operator on X. If f ∈ A+(T) and T is power-bounded operator,

then a bounded functional calculus naturally arises which can be dened by f(T ) :=∑∞k=0 f(k)T k ∈ L(X), where f(λ) =

∑∞k=0 f(k)λk and

∑∞k=0 |f(k)| <∞.

56

5.3 A KatznelsonTzafriri type theorem in Hilbert spaces

In Chapter 4, our starting point is an observation which leads us to introduce the

ergodic condition in our generalization.

Lemma. Let T be a power-bounded operator on a complex Banach space X and let

f ∈ A+(T). Then, for every λ ∈ T, we have

limn→∞

1n

∥∥∥∥∥n−1∑k=0


∥∥∥∥∥ = 0.

An important corollary of this lemma is the following statement which provides a link

between a scalar condition and an operator ergodic condition.

Corollary. Let T be a power-bounded operator on a Banach space X and f ∈ A+(T).Then, for each λ ∈ σ(T ) ∩ T,

f(λ) = 0 if and only if limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT kf(T )

∥∥∥∥∥ = 0.

The main result of Chapter 4 which applies to Hilbert spaces is the following:

Theorem. Let T be a power-bounded operator on a Hilbert space H. If Q ∈ L(H) and

TQ = QT, then the following statements are equivalent:

(i) limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT kQ

∥∥∥∥∥ = 0 for every λ ∈ σ(T ) ∩ T,


Moreover, if Q = f(T ) for some f ∈ A+(T), then (i) and (ii) are equivalent to

(iii) f(λ) = 0 for every λ ∈ σ(T ) ∩ T.

Many extensions of the KatznelsonTzafriri were proved in the discrete case as well

as in the continuous case; see [8], [24], [33], [51], [52], [65] and [4], [13]. However,

we recall that all former extensions of the KatznelsonTzafriri theorem are related to

bounded functional calculi of T or elements of the Banach algebra generated by T.

It remains an open problem whether this ergodic statement remains valid in more

general spaces; for instance, in Lp-spaces (1 < p < ∞) or in superreexive Banach

spaces. It is also an open problem whether one can prove a similar statement for C0-

semigroups or more general representations.

57

Chapter 6

Összefoglalás

Disszertációmban er®sen folytonos operátorfélcsoportok stabilitását vizsgáltam. Két

nevezetes stabilitási tétel az ArendtBattyLyubichVu és a KatznelsonTzafriri tétel

egy-egy kiterjesztését adtam meg, illetve karakterizáltam azokat a C0-félcsoportokat

is, amelyek normafüggvénye ún. topologikus regularitást mutat. A disszertációmban

szerepl® eredmények [42], [46] és [47] publikációkon alapulnak.

6.1 Lokálisan kompakt, kommutatív félcsoportok reprezen-

tációi

A disszertációm ezen része Kérchy Lászlóval közös cikken alapul.

Tegyük fel, hogy (G; +) egy lokálisan kompakt, σ-kompakt, kommutatív csoport.

Jelölje S egy olyan zárt részfélcsoportjátG-nek, amelynek S belseje nemüres, S−S = G

és S∩(−S) = 0 teljesülnek. A továbbiakban jelölje S karaktereinek a halmazát S]. A

2. fejezetben az S-en található invariáns közepeket használva bevezettük a normalizáló

függvény fogalmát, értelmeztük a reguláris norma viselkedés¶ reprezentációkat, illetve

vizsgáltuk a hozzájuk tartozó limeszfunkcionálok tulajdonságait.

Tétel. Legyen p normalizáló függvény S-en és tegyük fel, hogy létezik egy p-reguláris

normafüggvény¶ ρ : S → L(X) reprezentáció. Ekkor a p normalizáló függvény cp limesz-

funkcionálja S egy pozitív karaktere.

Tétel. Ha a ρ : S → L(X) reprezentáció reguláris norma viselkedés¶ a p és q norma-

lizáló függvényekre nézve, akkor

cp = cq.

58

6.1 Lokálisan kompakt, kommutatív félcsoportok reprezentációi

Az el®z® tétel lehet®vé teszi a következ® deniciót. A cρ := cp függvényt, amely

a normalizáló függvény megválasztásától független, a p-reguláris normafüggvény¶ ρ

reprezentáció limeszfunkcionáljának nevezzük. Beláttuk az S = R+ esetben, hogy egy

T : R+ → L(X) reguláris norma viselkedés¶ reprezentáció spektrálsugárfüggvénye meg-

egyezik T limeszfunkcionáljával, cT -vel.

Legyen f ∈ Cc(S) egy kompakt tartójú, folytonos függvény S-en. Az f függvény

ρ : S → L(X) reprezentáció szerinti Fourier transzformáltján az

f(ρ) :=∫Sf(s)ρ(s) dµ(s)

korlátos operátort értjük. Hasonlóan értelmezhetjük f(χ)-t tetsz®leges χ ∈ S]-re is,

hiszen S karakterei egy-dimenziós reprezentációk.

A 2. fejezetben bevezettük a nemkorlátos reprezentációk különféle spektrumfogal-

mait, részletesen megvizsgáltuk a köztük lév® kapcsolatot, illetve, hogy milyen rokon-

ságban állnak már ismert spektrumokkal.

A ρ reguláris normafüggvény¶ reprezentáció periferális spektrumán a

σ(ρ)per :=χ ∈ S] : |χ| = cρ és |f(χ)| ≤ ‖f(ρ)‖ minden f ∈ Cc(S)-re

halmazt értjük. Általában egy ρ : S → L(X) reprezentáció pontspektruma a

σp(ρ) :=χ ∈ S] : létezik 0 6= x ∈ X hogy ρ(s)x = χ(s)x minden s ∈ S-re

halmaz.

A következ® állítás a disszertáció egyik f® eredménye, amely egy ArendtBatty

LyubichVu típusú tétel reguláris normafüggvény¶ reprezentációkra. A pályák szokásos

stabilitása helyett egy gyengébb típus¶ stabilitást kapunk eredményül, amelyet az ún.

majdnem konvergencia fogalmával írhatunk le.

Tétel. Legyen a ρ : S → L(X) reprezentáció p-reguláris normafüggvény¶. Ha σper(ρ)

megszámlálható és σp(ρ∗) ∩ χ ∈ S] : |χ| = cρ üres, akkor ‖ρ(s)x‖/p(s) majdnem

konvergál zéróba minden x ∈ X-re.

A tétel következménye az alábbi állítás.

59

6.2 C0-félcsoportok és topologikus regularitás

Következmény. Legyen a ρ : S → L(X) reprezentáció p-reguláris norma-viselkedés¶.

Ha σper(ρ) megszámlálható és σp(ρ∗) ∩ χ ∈ S] : |χ| = cρ üres, akkor

limi→∞

1µ(Ki)

∫Ki

‖ρ(s)x‖p(s)

dµ(s) = 0

igaz minden x ∈ X-re, ahol Kii tetsz®leges Følner sorozat.

6.2 C0-félcsoportok és topologikus regularitás

A disszertáció 3. fejezetében a valós félegyenes reprezentációit vizsgáltuk meg részlete-

sebben. Itt az el®z® fejezet mintájára értelmeztük azokat a C0-félcsoportokat, amelyek

normafüggvénye topologikusan reguláris. Ez hasonlóan történt, mint a korábbi regula-

ritás esetében; a f® különbség, hogy deníciónkban az invariáns közepek halmaza helyett

egy sz¶kebb halmazt használtunk, a topologikusan invariáns közepeket.

A 3. fejezet f® eredménye megadja a topologikusan reguláris normafüggvénnyel ren-

delkez® C0-félcsoportok karakterizációját, hasonlóan a Kérchyt®l és Müllert®l származó

eredményhez [38], [43] a diszkrét esetben. Tegyük fel, hogy (T (s))s≥0 egy olyan C0-

félcsoport, ahol T (s) spektrálsugara pozitív, r(T (s)) > 0, minden s ∈ R+-re. Legyen

κT := infn∈N

sups∈R+

( 1n

∫ s+n

sr(T (t))−1‖T (t)‖ dt

)(sup

s≤y≤s+nr(T (y))−1‖T (y)‖

)−1 .

Jelölje a T -re nézve topologikus normalizáló függvények halmazát Pt. A következ® tétel

itt a f® eredményünk.

Tétel. Tekintsünk egy T : R+ → L(X) C0-félcsoportot. Ekkor az alábbi állítások ekvi-

valensek:

(i) létezik olyan p ∈ Pt, amelyre T normafüggvénye p-reguláris;

(ii) létezik olyan folytonos p ∈ Pt, amelyre T normafüggvénye p-reguláris;

(iii) ‖T (s)‖ ≥ 1 minden s ∈ R+-re és κT > 0.

60

6.3 KatznelsonTzafriri típusú tétel Hilbert tereken


Az ABLV tétellel rokonságot mutató operátorelméleti állítás a KatznelsonTzafriri tétel.

Az S = Z+ esetben megadtuk a tétel egy kiterjesztését Hilbert tereken. Célunk az

volt, hogy karaketrizáljuk a limn→∞ ‖TnQ‖ = 0 konvergenciát, ha T Hilbert térbeli

hatványkorlátos operátor és Q kommutál T -vel.

Jelölje I az identikus leképezést X-en. Ha f ∈ A+(T) és T ∈ L(X) hatványkorlátos

operátor, akkor értelmezhet® f(T ) :=∑∞

k=0 f(k)T k ∈ L(X), ahol f(λ) =∑∞

k=0 f(k)λk

és∑∞

k=0 |f(k)| < ∞. A 4. fejezet kiindulási pontja az alábbi észrevétel, amelyet egy

lemmában fogalmazunk meg.

Lemma. Legyen T hatványkorlátos operátor az X komplex Banach téren és legyen

f ∈ A+(T). Ekkor minden λ ∈ T esetén a következ® teljesül

limn→∞

1n

∥∥∥∥∥n−1∑k=0


∥∥∥∥∥ = 0.

A lemma fontos következménye az alábbi állítás.

Következmény. Legyen T hatványkorlátos operátor az X Banach téren és f ∈ A+(T).Ekkor tetsz®leges λ ∈ σ(T ) ∩ T esetén,

f(λ) = 0 akkor és csak akkor, ha limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT kf(T )

∥∥∥∥∥ = 0.

A 4. fejezet f® eredménye, amely Hilbert terekre vonatkozik a következ®:

Tétel. Legyen T hatványkorlátos operátor a H Hilbert téren. Ha Q ∈ L(H) és TQ =

QT, akkor a következ® állítások ekvivalensek:

(i) limn→∞

1n

∥∥∥∥∥n−1∑k=0

λ−kT kQ

∥∥∥∥∥ = 0 minden λ ∈ σ(T ) ∩ T-re,


Továbbá, ha Q = f(T ) valamely f ∈ A+(T)-re, akkor (i) és (ii) ekvivalens az alábbival

(iii) f(λ) = 0 minden λ ∈ σ(T ) ∩ T-re.

61


A KatznelsonTzafriri tételnek számos általánosítása született diszkrét és folytonos

reprezentációkra, ld. [8], [24], [33], [51], [52], [65] és [13], [4]. Megjegyezzük azonban,

hogy ezek az általánosítások mind T egy függvénykalkulusához kapcsolódnak vagy a T

által generált Banach algebrához.

Nyitott kérdés maradt, hogy az el®z® ergodikus állításunk érvényes-e Lp-terekben

(1 < p < ∞), vagy általánosabban, szuperreexív Banach terekben. További nyitott

probléma, hogy milyen hasonló állítás bizonyítható C0-félcsoportokra, illetve általáno-

sabb reprezentációkra.

62

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On Stability Conditions of Operator Semigroups · semigroups we refer the reader to Engel Nagel's monograph [20] and [2]. One of the most studied parts of the theory of linear operator