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RESULTS RELATED TO THEOREMS OFKATZNELSON-TZAFRIRI TYPE FOR REPRESENTATIONS

OF SEMIGROUPS.

M. ZARRABI

Abstract. Let S be a suitable semigroup of a locally compact abelian

group. If T is a representation of S by contractions on a Hilbert space,

we show that inft∈S ‖T (t) bf(T )‖ = sup {|f(λ)|, λ ∈ Spu(T, S)}, where

Spu(T, S) is the unitary spectrum of T with respect to S. When T

is a representation of S by contractions on a Banach space, we give

sharp conditions on Spu(T, S) which guarantee that the equality above

holds. These conditions concern the thinness of Spu(T, S), in the har-

monic analysis sense. These results are related to theorems of Katzelson-

Tzafriri type, which give conditions guaranteeing that inft∈S ‖T (t) bf(T )‖vanishes.

1. Introduction

Let G be a locally compact abelian group equipped with the Haar measureand S be a suitable measurable subsemigroup of G such that G = S − S.Let T be a representation of G by contractions on a Banach space X. Forf ∈ L1(S), we set

f(T ) : x 7−→∫Sf(t)T (t)xdt, x ∈ X.

The subsemigroup S will be ordered in the following way: s 4 t⇔ t−s ∈ S.So we shall denote limS , the limit as t→∞ through S. The purpose of thispaper is to study the asymptotic behaviour of ‖T (t)f(T )‖, as t→∞. Thisis related to Theorems of Katznelson-Tzafriri type ([17],[10],[11],[20],[3]) andrecent results of [19]. The unitary spectrum of T with respect to S is definedby

Spu(T, S) = {χ ∈ Γ; |f(χ)| ≤ ‖f(T )‖ for all f in L1(S)},where Γ is the dual group of G. It is shown by Batty and Phong in[3], that if f ∈ L1(S) satisfies the spectral synthesis for Spu(T, S), thenlimS ‖T (t)f(T )‖ = 0. This result was first proved by Katznelson and Tzafriri

2000 Mathematics Subject Classification. 47D03; 43A46.

Key words and phrases. representations, Locally compact abelian group, semigroup,

contractions, von Neumann inequality, spectral synthesis, Helson set.

1

2 M. ZARRABI

([17]) in the case S = Z+. More precisely, they showed that if T1 is a con-traction on a Banach space and if h is an analytic function on the unitdisk D, which has absolutely convergent Taylor series and satisfies spectralsynthesis for σ(T1) ∩ T, then

limn→+∞

‖Tn1 h(T1)‖ = 0. (1.1)

Here σ(T1) denote the spectrum of T1 and T the unit circle. When T1 is acontraction on a Hilbert space, Esterle, Strouse and Zouakia showed in [10]that (1.1) holds for every function h in the disk algebra which vanishes onσ(T1) ∩ T.

The case of C0-semigroups was also studied. It was shown independentlyin [11] and [20] that if T = (T (t))t≥0 is a C0-semigroups, with generator A,then for every f ∈ L1(R+) which satisfies spectral synthesis for iσ(A) ∩ R,limt→+∞ ‖T (t)f(T )‖ = 0.

On the other hand Mustafaev showed ([19]) that if T = (T (t))t≥0 is anorm-continuous semigroup of contractions on a Banach space with genera-tor A (which is a bounded operator), then

limt→∞‖T (t)A‖ = sup {|λ|, λ ∈ σ(A) ∩ iR} .

He proved also that if σ(A) ∩ iR ⊂ [−iπ/2, iπ/2],

limt→∞‖T (t) sinA‖ = sup {| sinλ|, λ ∈ σ(A) ∩ iR} .

If F is a topological space, h a continous bounded function on E andE ⊂ F , we set ‖h‖C(E) = sup {|f(λ)|, λ ∈ E}, with the understanding that‖h‖C(E) = 0 if E is empty. Let T be a representation by contractions on aHilbert space. We assume that an equivalent version of the J. von Neumanninequality holds for T , that is for every f ∈ L1(S), ‖f(T )‖ ≤ ‖f‖C(Γ). Weshow then (Theorem 2.5) that for every f ∈ L1(S),

limS‖T (t)f(T )‖ = ‖f‖C(Spu(T,S)). (1.2)

As a consequence we obtain the following results:(a) If T = (T (t))t≥0 is a C0-semigroup of contractions on a Hilbert space

with generator A, then for every f ∈ L1(R+),

limt→+∞

‖T (t)f(T )‖ = ‖f‖C(iσ(A)∩R).

(b) If T1 is a contraction on a Hilbert space then for every function h inthe disk algebra, we have limn→+∞ ‖Tn1 h(T1)‖ = ‖h‖C(σ(T1)∩T).

(c) If T1 and T2 are two commuting contractions on a Hilbert space andh is a holomorphic function on D2 and continuous on D2, then

limmin (n1,n2)→+∞

‖Tn11 Tn2

2 h(T1, T2)‖ = ‖h‖C(σA(T1,T2)∩T2),

REPRESENTATIONS OF ABELIAN SEMIGROUPS 3

where A is any unitary commutative Banach algebra that contains T1, T2

and σA(T1, T2) is the joint spectrum of T1 and T2 relative to A.We do not know if the last result extends to three commuting contractions.

Indeed the corresponding version of the J. von Neumann inequality holdsfor two commuting contractions, while it fails for three contractions.

These results are far from true for representations by contractions actingon a Banach space. As we will see equality (1.2) depends on the thinnessof Spu(T, S). Notice that we always have ‖f‖C(Spu(T,S)) ≤ limS ‖T (t)f(T )‖.Let E be a closed subset of Γ. We shall now denote by T a representationof S by contractions on a Banach space. We show (Corollary 5.5) thatequality (1.2) holds for every T such that Spu(T, S) = E, if and only if Esatisfies spectral synthesis and is a Helson set with α(E) = 1, where α(E)is the Helson constant of E. We obtain this as a consequence of a moregeneral result (Theorem 5.4), from which we deduce that if E ⊂ T satisfiesspectral synthesis and is a Helson set then there exists a constant c suchthat limn→+∞ ‖Tn1 h(T1)‖ ≤ c‖h‖C(E), for every contraction T1 such thatσ(T1) ∩ T = E. So we investigate in section 6 conditions on E for which wemay replace in the above inequality the supremum norm by a bigger one,namely the norm

‖h‖Λϕ(E) := ‖h‖C(E) + sup{|h(z)− h(ζ)|ϕ(|z − ζ|)

, z, ζ ∈ E and z 6= ζ

},

where ϕ : R+ 7→ R+ is a nondecreasing function, with ϕ(0+) = 0 and suchthat the function t → ϕ(t)

t is nonincreasing. We denote by Λϕ(E) the setof all continuous functions h on E such that ‖h‖Λϕ(E) is finite and by A(T)the Wiener algebra of continuous functions which has absolutely convergentTaylor series. We set

A+(T) ={h ∈ A(T) : h(n) = 0, n < 0

}.

Let Et = {eis ∈ T, ∃ eiu ∈ E such that |s − u| ≤ t}, t > 0 and |Et| theLebegue measure of |Et|. We show (Theorem 6.1) that if∫ 1

0ϕ(t)|Et|1/2t−3/2dt < +∞, (1.3)

then, there exists a constant c (depending only on E and ϕ) such that

limn→+∞

‖Tn1 h(T1)‖ ≤ c‖h‖Λϕ(E),

for every contraction T1 with σ(T1) ∩ T = E, and h ∈ A+(T) ∩ Λ+ϕ (T).

We also show (Theorem 6.2) that under an additional regularity conditionon ϕ, condition (1.3) is sharp when E is a perfect symetric set of constantratio. These results are closely linked to results concerning the inclusion

4 M. ZARRABI

Λϕ(E) ⊂ A(E) := {h|E , h ∈ A(T)}, where the condition (1.3) appears (see[15], p. 133).

2. Representations of subsemigroups by contractions on a

Hilbert space

We shall adopt the terminology of [2] and [3]. Let G be a locally compactabelian group and S be a measurable subsemigroup of G with non emptyinterior in G and such that G = S − S. The group G is equipped withthe Haar measure and S with the restriction of that measure. L1(S) willbe identified with a subspace of L1(G). Denote by S∗ the set of all thenon-zero, continuous, bounded, homomorphisms of S into the multiplicativesemigroup C. Let

S∗u = {χ ∈ S∗; |χ(s)| = 1 for all s in S}.

We shall identify S∗u with the dual group Γ of G, in the natural way. Forf ∈ L1(S) and χ ∈ S∗, we set

f(χ) =∫Sf(t)χ(t)dt.

Finally we shall assume that {f , f ∈ L1(S)} separates the points of S∗ fromeach other and from zero.

Exemples. 1) Let G = Rn, S = Rn+. The set S∗ will be identified with

C−n, where C− = {z ∈ C : Im(z) < 0}. The identification map is given by

z → χz, where χz(t) = e−it.z, z = (z1, . . . , zn) ∈ C−n, t = (t1, . . . , tn) ∈ Rn

+

and t.z = t1z1 + . . .+ tnzn. For f ∈ L1(Rn+) and z ∈ C−

n, we have

f(z) = f(χz) =∫

Rn+f(t)e−it.zdt.

Clearly S∗u = Rn.2) Let G = Zn and S = Zn+, where Z+ is the set of all nonnegative

integers; S∗ will be identified with Dn by the map z = (z1, . . . , zn) → χz,where χz(k) = zk1

1 . . . zknn and k = (k1, . . . , kn) ∈ Zn+. We have S∗u = Tn and,for f ∈ L1(Zn+),

f(z) =∑k∈Zn+

f(k)zk11 . . . zknn .

Let X be a Banach space, L(X) the Banach algebra of all bounded linearoperators on X. Let T be a representation of S by contractions on X, thatis a strongly continuous homomorphism from S into L(X) such that T (s)is a contraction whenever s ∈ S. For f ∈ L1(S), let f(T ) : X 7−→ X be the


bounded operator defined by

f(T ) : x 7−→∫Sf(t)T (t)xdt.

The spectrum of T with respect to S is defined by

Sp(T, S) = {χ ∈ S∗; |f(χ)| ≤ ‖f(T )‖ for all f in L1(S)}.

We shall mainly be interested in the unitary part of the spectrum of T ,which is defined by Spu(T, S) = Sp(T, S) ∩ S∗u = Sp(T, S) ∩ Γ.

Remark. We recall here a result that we shall need about the structureof closed ideals in the algebra of continuous functions on a locally compactHausdorff space Z. Denote by C0(Z) the algebra of all continuous functionson Z vanishing at infinity. As usual, C0(Z) is endowed with the supremumnorm. When Z is compact, C0(Z) = C(Z) and it is well known that everyclosed ideal I of C0(Z) has the form I = I(E), where E is a closed subsetof Z and I(E) = {h ∈ C0(Z), h = 0 on E} [24] (see also [12], Corollary, p.204).

It is easily seen that this result remains true when Z is not compact. Thisfollows from the above result and from the fact that every closed ideal ofC0(Z) is also a closed ideal of C(Z∞), where Z∞ = Z∪{∞} is the Alexandroffcompactification of Y .

Since C0(Z) is a regular Banach algebra (see [16], for definition and prop-erties), the result above is equivalent to the fact that for any closed subsetE of Z, we have J (E) = I(E), where J (E) is the closure in C0(Z) of theset of all functions vanishing in a neighbohood of E.

Notice also that if E ⊂ Z is closed then C0(E) is isometrically isomorphicto the quotient Banach algebra C0(Z)/I(E), which means that ‖f‖C(E) =infh∈I(E) ‖f − h‖C(Z).

We need the following version of the Douglas lemma ([6]) given in [2].

Lemma 2.1. Let V be a representation of S by isometries on a Banachspace Y . There exists a Banach space Yd containing Y (by isometric iso-morphism), and a representation Vd of G by isometries on Yd such thatVd(s)y = V (s)y (y ∈ Y, s ∈ S) and Sp(Vd, G) = Spu(V, S).

Lemma 2.2. Let V be a representation of S by isometries on a Banachspace Y and y ∈ Y . Assume that for every f ∈ L1(S),

‖f(V )y‖ ≤ ‖f‖C(Γ).

Then, for every f ∈ L1(S),

‖f(V )y‖ ≤ ‖f‖C(Spu(V,S)).

6 M. ZARRABI

Proof. Let Vd be the representation of G by isometries on a Banach spaceYd, given by Lemma 2.1.

We assume that Y 6= {0}. By [2], Corollary 2.2, Spu(T, S) is non-empty.Let ε > 0 and g ∈ L1(G). Since S has a nonempty interior, there existss ∈ S such that ∫

G\S−s|g(t)|dt < ε.

Thus

‖g(Vd)y‖ ≤∥∥∥∥∫

S−sg(t)Vd(t)ydt

∥∥∥∥+

∥∥∥∥∥∫G\S−s

g(t)Vd(t)ydt

∥∥∥∥∥≤∥∥∥∥∫

Sg(t− s)Vd(t− s)ydt

∥∥∥∥+ ε‖y‖

≤∥∥∥∥Vd(−s)(∫

Sg(t− s)Vd(t)ydt

)∥∥∥∥+ ε‖y‖

≤∥∥∥∥∫

Sgs(t)Vd(t)ydt

∥∥∥∥+ ε‖y‖ = ‖gs(Vd)y‖+ ε‖y‖,

where

gs(t) ={g(t− s) if t ∈ S

0 otherwise.

Hence

‖gs(Vd)y‖ = ‖gs(V )y‖ ≤ ‖gs‖C(Γ).

On the other hand, for every χ ∈ Γ, we have

gs(χ) = g(χ)−∫G\S−s

g(t)χ(t)dt,

which implies that ‖gs‖C(Γ) ≤ ‖g‖C(Γ) + ε. Therefore, we get

‖g(Vd)y‖ ≤ ‖g‖C(Γ) + ε(1 + ‖y‖).

Letting ε→ 0, we obtain

‖g(Vd)y‖ ≤ ‖g‖C(Γ).

Let f ∈ L1(S) and let g ∈ L1(G) be such that g vanishes on a neighborhoodof Spu(V, S). Since Spu(V, S) = Sp(Vd, G), we have g(Vd) = 0 (see [18], p.201). It follows then from the last inequality that

‖f(V )y‖ ≤ ‖f(Vd)y − g(Vd)y‖ ≤ ‖f − g‖C(Γ).

Since A(Γ) = {g, g ∈ L1(G)} is dense in C0(Γ) (see [21], Theorem 1.2.4,p. 9), it follows from the structure of closed ideals of C0(Γ) recalled above


that {g, g ∈ L1(G) and g = 0 on a neighborhood of Spu(V, S)} is dense inI (Spu(V, S)) (= {h ∈ C0(Γ), h = 0 on Spu(V, S)}). Hence

‖f(V )y‖ ≤ infh∈I(Spu(V,S))

‖f − h‖C(Γ) = ‖f‖C(Spu(V,S)).

�

The following is Lemma 4.1 of [20].

Lemma 2.3. Let T be a representation of S by contractions on a Banachspace X. Then there exist a Banach space Y , a continuous homomorphismπ : X 7→ Y with dense range, and a representation V of S by isometries onY , such that

(i) Sp(V, S) ⊂ Sp(T, S),

(ii) V (t) ◦ π = π ◦ T (t) for every t in S,

(iii) limS‖T (t)x‖ = ‖πx‖ for every x in X.

Lemma 2.4. Let T be a representation of S by contractions on a Banachspace X. Let x ∈ X. Assume that for every f ∈ L1(S),

limS‖T (t)f(T )x‖ ≤ ‖f‖C(Γ).

Then, for every f ∈ L1(S),

limS‖T (t)f(T )x‖ ≤ ‖f‖C(Spu(T,S)).

Proof. Let V and π be respectively, the isometry and the contractive homo-morphism given by Lemma 2.3. Let f ∈ L1(S). Applying (ii) and then (iii)of Lemma 2.3, we obtain

‖f(V )πx‖ = ‖π(f(T )x)‖ = limS‖T (t)f(T )x‖.

Then we get ‖f(V )πx‖ ≤ ‖f‖C(Γ). It follows from Lemma 2.2 that

limS‖T (t)f(T )x‖ = ‖f(V )πx‖ ≤ ‖f‖C(Spu(V,S)) ≤ ‖f‖C(Spu(T,S)).

The last inequality holds since Spu(V, S) ⊂ Spu(T, S). �

Theorem 2.5. Let T be a representation of S by contractions on a Banachspace X. Assume that for every f ∈ L1(S),

limS‖T (t)f(T )‖ ≤ ‖f‖C(Γ).

Then for every f ∈ L1(S),

limS‖T (t)f(T )‖ = ‖f‖C(Spu(T,S)).

8 M. ZARRABI

Proof. Let f ∈ L1(S). For t ∈ S, we have

T (t)f(T ) = ft(T )

where

ft(s) ={f(s− t) if s ∈ t+ S

0 otherwise.

Let χ ∈ Spu(T, S). We clearly have the following equality ft(χ) = χ(t)f(χ).Thus, using the inequality |ft(χ)| ≤ ‖ft(T )‖, we get |f(χ)| ≤ ‖T (t)f(T )‖.So ‖f‖C(Spu(T,S)) ≤ limS ‖T (t)f(T )‖.

On the other hand, for t ∈ S, let M(t) be the linear bounded operatordefined on L(X) by

M(t) : A 7−→ T (t)A.

Then M is a representation of S by contractions on L(X). Moreover, forf ∈ L1(S), f(M) is the operator on L(X) defined by

f(M) : A 7−→ f(T )A.

Clearly, we have Sp(M,S) = Sp(T, S) and Spu(M,S) = Spu(T, S). More-over, we have limS ‖M(t)f(M)I‖ = limS ‖T (t)f(T )‖ ≤ ‖f‖C(Γ), where I isthe identity map on X. Applying now Lemma 2.4 to M and I, we obtain

limS‖T (t)f(T )‖ = lim

S‖M(t)f(M)I‖ ≤ ‖f‖C(Spu(T,S)).

�

3. C0-semigroups and contractions

Here we consider the case G = R and S = R+. We recall (see section2) that S∗ = C−, S∗ = Γ = R, and for z ∈ C− and f ∈ L1(R+), we havef(z) = f(χz) =

∫ +∞0 f(t)e−itzdt.

Let T = (Tt)t≥0 be a strongly continuous semigroup of contractions on aHilbert space H and A be its generator. Let x ∈ Spu(T,R+) and , for ε > 0,fε(t) = e(ix−ε)t, t ≥ 0. We have fε(x) = 1

ε and fε(T ) = (−ix + ε − A)−1.Since |fε(x)| ≤ ‖fε(T )‖, we get 1

ε ≤ ‖(−ix+ε−A)−1‖. As the last inequalityholds for every ε > 0, we see that −ix ∈ σ(A).

On the other hand if −ix ∈ σ(A) ∩ iR, we have f(x) ∈ σ(f(T )) for everyf ∈ L1(R+) and so x ∈ Spu(T,R+). We deduce that Spu(T,R+) = iσ(A)∩R.

Theorem 3.1. Let T = (T (t))t≥0 be a strongly continuous semigroup ofcontractions on a Hilbert space H and, let A be its generator. Then forevery f ∈ L1(R+),

limt→+∞

‖T (t)f(T )‖ = ‖f‖C(iσ(A)∩R).


Proof. Denote by D the domain where A is defined. Let v ∈ D, with ‖v‖ = 1.We have

〈Av, v〉 = limt→+∞

⟨1t(T (t)v − v), v

⟩= lim

t→+∞

1t

(〈T (t)v, v〉 − 1) ,

where 〈., .〉 is the scalar product of H. Since |〈T (t)v, v〉| ≤ 1, we haveRe (〈Av, v〉) ≤ 0. So the numerical range {〈−Av, v〉, ‖v‖ = 1}, of −A iscontained in −iC+. It follows from the von Neumann inequality (see [5])that for every rationnal functions r bounded on −iC+ and with poles iniC+, we have

‖r(−A)‖ ≤ supz∈−iC+

|r(z)| = supz∈iR|r(z)|. (3.1)

We denote by F the set of all functions of the form

f(t) =n∑k=1

akeλkt, (3.2)

where (ak)1≤k≤n and (λk)1≤k≤n are a complex numbers such that for everyk, λk is of a negative real part. For f in F of the form (3.2), we have

f(z) =∫ +∞

0f(t)e−iztdt =

n∑k=1

ak(iz − λk)−1, z ∈ C−.

and

f(T ) =n∑k=1

ak(−A− λk)−1 = r(−A),

where r(z) = f(−iz) =∑n

k=1 ak(z−λk)−1, z ∈ −iC+. It follows from (3.1)that

‖f(T )‖ ≤ supz∈iR|r(z)| = sup

x∈R|f(x)|

Since F is dense in L1(R+), the last inequality holds for every function f inL1(R+). Now we apply Theorem 2.5 to finish the proof.

�

Let λ ∈ C, with positive real part. Under the hypothesis of the Theorem3.1, we get limt→+∞ ‖T (t)(λI − A)−1‖ = 1

dist(λ,σ(A)∩iR) . More generallyif a0, . . . , an are complex numbers, applying the above theorem to f(t) =(∑n

k=0akk! t

k)e−λt, we obtain

limt→+∞

∥∥∥∥∥T (t)n∑k=0

ak(λI −A)−k−1

∥∥∥∥∥ = supx∈iσ(A)∩R

∣∣∣∣∣n∑k=0

ak(λ+ ix)−k−1

∣∣∣∣∣= sup

z∈σ(A)∩iR

∣∣∣∣∣n∑k=0

ak(λ− z)−k−1

∣∣∣∣∣ .

10 M. ZARRABI

Now we turn to the case G = Z and S = Z+. We have S∗ = D andS∗u = Γ = T. Every representation T of Z+ by contractions on H is givenby T = (Tn1 )n∈Z+ , where T1 is a contraction on H.

Let λ ∈ D and f = (f(n))n∈Z+ ∈ `1(Z+). We have f(λ) =∑

n≥0 f(n)λn

and f(T ) = f(T1) =∑

n≥0 f(n)Tn.If λ ∈ σ(T1), then f(λ) ∈ σ(f(T1)), which implies that |f(λ)| ≤ ‖f(T1))‖.

Thus σ(T1) ⊂ Sp(T,Z+).Let λ ∈ Spu(T,Z+). Then, for every r > 1, we have |fr(λ)| ≤ ‖fr(T )‖,

where fr(z) = (rλ − z)−1. So, for every r > 1, (r − 1)−1 ≤ ‖(rλ − T1)−1‖,which implies that λ ∈ σ(T1) ∩ T. It follows that Spu(T,Z+) = σ(T1) ∩ T.

Theorem 3.2. Let T1 be a contraction on a Hilbert space H. Then forevery h in the disk algebra A(D),

limn→+∞

‖Tn1 h(T1)‖ = ‖h‖C(σ(T1)∩T).

Proof. Let f = (f(n))n≥0 ∈ `1(Z+). By the J. von Neumann inequality, wehave ‖f(T )‖ ≤ ‖f‖C(T), where T = (Tn1 )n∈Z+ . Then we get from Theorem2.5 that limn→+∞ ‖Tn1 f(T1)‖ = ‖f‖C(Spu(T,Z+)). The equality in the theoremfollows since Spu(T,Z+) = σ(T1) ∩ T and since the set {f , f ∈ `1(Z+)} isdense in A(D). �

As consequence we obtain the following curious result which characterizesthe peripheral spectrum of a contraction on a Hilbert space. The correspond-ing result for contractions acting on a Banach space fails.

Corollary 3.3. Let T1 be a contraction on a Hilbert space H and λ ∈ D.Then λ ∈ σ(T1) ∩ T if and only if limn→+∞ ‖Tn1 (λI + T1)‖ = 2.

Proof. By Theorem 3.2, we have limn→+∞ ‖Tn1 (λI+T1)‖ = supz∈σ(T1)∩T |λ+z|. On the other hand, for λ ∈ D, supz∈σ(T1)∩T |λ+ z| = 2 holds if and onlyif λ ∈ σ(T1) ∩ T, which finishes the proof. �

This result fails for contractions on a Banach space. Consider the spaceC2, endowed with the norm ‖(z1, z2)‖ = max{|z1|, |z2|} and the operatordefined on C2 by the matrix

T1 =(

0 11 0

).

Clearly T1 is an isometry, σ(T1) = {1,−1} and for every λ ∈ T, we havelimn→+∞ ‖Tn1 (λI + T1)‖ = ‖λI + T1‖ = 2.


4. Several commuting contractions

Let k ≥ 2 be an integer. We shall write n = (n1, . . . , nk) ∈ Zk+. LetT be a representation of Zk+ by contractions on a Hilbert space H. Wehave T = (Tn1

1 . . . Tnkk )n∈Zk+, where T1, . . . , Tk are finite many commuting

contractions on H.There are several definitions of the joint spectrum. Let A be a commu-

tative Banach algebra that contains T1, . . . , Tk and I the identity map. Wedefine the joint spectrum σA(T1, . . . , Tk), relative to A, to be the set of allk-tuples of complex numbers λ = (λ1, . . . , λk) such that the ideal (and/orthe closed ideal) of A generated by λ1 − T1, . . . , λk − Tk is a proper subsetof A.

Let f = (f(n))n∈Zk+in `1(Zk+). We have

f(λ) =∑n∈Zk+

f(n)λn11 . . . λnkk , λ = (λ1, . . . , λk) ∈ Dk

,

and

f(T ) =∑n∈Zk+

f(n)Tn11 . . . Tnkk .

Let λ ∈ σA(T1, . . . Tk). It is easily seen that f(T ) − f(λ) is contained inthe proper closed ideal generated by λ1−T1, . . . , λk−Tk. So f(λ) ∈ σ(f(T )),which implies in particular that |f(λ)| ≤ ‖f(T )‖. Thus σA(T1, . . . , Tk) ⊂Sp(T,Zk+).

Let now λ ∈ Spu(T,Zk+). Since Sp(T,Zk+) is σ-compact, λ is an ω-approximate eigenvalue of T , that is, there exists a sequence (xp)p in H

with ‖xp‖ = 1 and such that ‖Tn11 . . . Tnkk xp − λn1

1 . . . λnkk xp‖ → 0, asp → +∞, for each n ∈ Zk+ (see [3], Proposition 2.2). In particular, forevery i = 0, . . . , k, ‖(Ti − λi)xp‖ → 0 as p→ +∞. It follows that the equa-tion (T1 − λ1)S1 + . . . + (Tk − λk)Sk = I has no solution S1, . . . , Sk ∈ A.Thus λ ∈ σA(T1, . . . , Tk).

We deduce from the above observations that σA(T1, . . . Tk)∩Tn = Spu(T,Zk+).We see that σA(T1, . . . , Tk) ∩ Tn is the same for any commutative Banachalgebra that contain T1, . . . , Tk and I.

Theorem 4.1. Let T1 and T2 be two commuting contractions on H. Thenfor every function h holomorphic in D2 and continuous on D2, we have

limmin (n1,n2)→+∞

‖Tn11 Tn2

2 h(T1, T2)‖ = ‖h‖C(σA(T1,T2)∩T2),

where A is any commutative Banach algebra that contains T1, T2 and theidentity map.

12 M. ZARRABI

Proof. The corresponding version of the J. von Neumann inequality fortwo commuting contractions is also true, that is for any polynomial p ∈C[z1, z2], ‖p(T1, T2)‖ ≤ ‖p‖C(T2). This is a consequence of the Ando result[1]. The proof follows now from Theorem 2.5, the fact that σA(T1, T2)∩T2 =Spu(T,Z2

+) and the density of C[z1, z2] in the algebra of holomorphic func-tions in D2 and continuous on D2. �

Remarks. 1. We do not know if Theorem 4.1 can be extended to three(or more) commuting contractions. Notice that the corresponding version ofthe J. von Neumann inequality fails for three commuting contractions (see[4], [7]).

2. Let T1, . . . , Tk be finitely many commuting contractions on H. Assumethat one of the following conditions holds:

(i) H is two dimensionnal.(ii) Every Ti commutes with every T ∗j .(iii)

∑ni=1 ‖Ti‖2 ≤ 1.

Then

‖Tn11 Tn2

2 . . . Tnkk h(T1, . . . , Tk)‖ → ‖h‖C(σA(T1,...,Tk)∩Tk),

as min(n1, . . . , nk) → +∞. Indeed under one of the above conditions (i) −(iii), the J. von Neumann inequality holds for T1, . . . , Tk (see [7]).

5. Representations of subsemigroups by contractions on a

Banach space

Let G, Γ and S be as in section 2. If I is a closed ideal of L1(G), we set

h(I) = {χ ∈ Γ : f(χ) = 0, for all f ∈ I},

where f(χ) =∫G f(t)χ(t)dt.

Let E be a closed subset of Γ. We set

I(E) = {f ∈ L1(G), f = 0 on E}

andJ(E) = clos {f ∈ L1(G), f = 0 on a neighborhood of E}.

Clearly I(E) and J(E) are closed ideals of L1(G), J(E) ⊂ I(E) and wehave h(I(E)) = h(J(E)) = E (see [21], p. 161). The set E is said to satisfyspectral synthesis if J(E) = I(E). Notice that E satisfies spectral synthesisif there exists a unique closed ideal I such that h(I) = E. A functionf ∈ L1(G) (or f) is said to satisfy spectral synthesis for E if f ∈ J(E).

Let A(Γ) denote the Banach algebra of functions f , f ∈ L1(G) endowedwith the norm ‖f‖A(Γ) = ‖f‖L1(G). Let A(E) = {f|E , f ∈ L1(G)} bethe Banach space equipped with the quotient norm of L1(G)/I(E), that is‖f‖A(E) = infg∈I(E) ‖f − g‖L1(G). It is well known that A(E) ⊂ C0(E). The


set E is called a Helson set if A(E) = C0(E). Notice that E is a Helsonset if and only if the canonical injection i : A(E) → C0(E) is onto. This isequivalent to the fact that the quantity

α(E) = sup

{‖f‖A(E)

‖f‖C(E)

, f ∈ L1(G) and f|E 6= 0

},

is finite. By duality, we have

α(E) = sup{‖µ‖‖µ‖PM

, µ ∈M(E) and µ 6= 0},

where M(E) is the set of Borel measures on Γ with support contained in E,‖µ‖ is the total variation of µ and ‖µ‖PM = supx∈G |µ(x)|; α(E) is calledthe Helson constant of E (see [13]).

Lemma 5.1. Let V be a representation of S by isometries on a Banachspace Y and y ∈ Y . Assume that Spu(V, S) ⊂ E, where E ⊂ Γ is a closedset that satisfies spectral synthesis and which is a Helson set. Then for everyf ∈ L1(S),

‖f(V )y‖ ≤ α(E)‖f‖C(E)‖y‖.

Proof. Let Vd be as in Lemma 2.1, with Sp(Vd, G) = Spu(V, S). Let f ∈L1(S). Since E satisfy spectral synthesis, for every g ∈ L1(G) such that gvanishes on E, we have g(Vd) = 0 (see [18], p. 201). So

‖f(V )y‖ = ‖f(Vd)y‖ ≤ infg∈I(E)

‖f(Vd)y − g(Vd)y‖

≤ infg∈I(E)

‖f − g‖‖y‖ = ‖f‖A(E)‖y‖.

Now since E is a Helson set, we get ‖f(V )y‖ ≤ α(E)‖f‖C(E)‖y‖.�

Lemma 5.2. Let T be a representation of S by contractions on a Banachspace X and x ∈ X. Assume that E := Spu(T, S) satisfies spectral synthesisand is a Helson set. Then for every f ∈ L1(S),

limS‖T (t)f(T )x‖ ≤ α(E)‖f‖C(E)‖x‖.

Proof. Let V and π be respectively, the isometry and the contractive homo-morphism given by Lemma 2.3. Let f ∈ L1(S). We have limS ‖T (t)f(T )x‖ =‖f(V )πx‖ and by Lemma 5.1, we get

limS‖T (t)f(T )x‖ ≤ α(E)‖f‖C(E)‖πx‖ ≤ α(E)‖f‖C(E)‖x‖.

�

We shall also need the following lemma.

14 M. ZARRABI

Lemma 5.3. Let T be a representation of G by isometries on a Banachspace. Let I = {f ∈ L1(G) : f(T ) = 0}. Then I is a closed ideal of L1(G)and Spu(T, S) = Sp(T,G) = h(I).

Proof. It is easily seen that I is a closed ideal. The inclusion Sp(T,G) ⊂Spu(T, S) and Sp(T,G) ⊂ h(I) are straightforward.

Let χ ∈ Spu(T, S), f ∈ L1(G) and ε > 0. There exists s ∈ S such that∫G\(S−s) |f(t)|dt < ε. We set

fs(t) ={f(t− s) if t ∈ S

0 otherwise.

We have fs ∈ L1(S) and

|f(χ)| ≤ |fs(χ)|+ ε ≤ ‖fs(T )‖+ ε ≤ ‖f(T )‖+ 2ε.

Hence |f(χ)| ≤ ‖f(T )‖, for every f ∈ L1(S) and thus χ ∈ Sp(T,G). There-fore Sp(T,G) = Spu(T, S).

On the other hand if f ∈ J(Sp(T,G), we have f(T ) = 0 ([18], p. 201).Thus J(Sp(T,G) ⊂ I and so h(I) ⊂ Sp(T,G), which finishes the proof.

�

Theorem 5.4. Let c ≥ 1 and E a closed subset of Γ. Then the followingare equivalents.

(i) E satisfies spectral synthesis and is a Helson set with α(E) ≤ c.(ii) For every representation T of S by contractions on a Banach space

X such that Spu(T, S) = E, we have

limS‖T (t)f(T )‖ ≤ c‖f‖C(E), f ∈ L1(S).

Proof. (i)⇒ (ii) Let T be a representation of S by contractions on X withSpu(T, S) = E. Let M(t) : A 7−→ T (t)A; M is a representation of S bycontractions on L(X) with Spu(M,S) = E, and for every f ∈ L1(S), wehave f(M) : A 7−→ f(T )A. It suffices now to apply Lemma 5.2 to M andthe identity map I.

(ii) ⇒ (i) Let I be a closed ideal of L1(G) with h(I) = E. Let π :L1(G) → L1(G)/I denote the canonical surjection and, for t ∈ G, τt thetranslation operator defined on L1(G) by : τtf(x) = f(x − t), x ∈ G. Weset T (t)(π(f)) = π(τtf), f ∈ L1(G). For every t, T (t) and T (t)−1 = T (−t)are contractions, which means that T (t) is an invertible isometry.

For f ∈ L1(G), f(T ) is the operator defined by: π(g)→ π(f ∗ g). Hencef(T ) = 0 holds if and only if f ∗ g ∈ I, whenever g ∈ L1(G). Since thereexists a sequence (gn)n in L1(G) such that ‖f − f ∗ gn‖ → 0 (see [21],Proposition 2.6.6), we see that f(T ) = 0 holds if and only if f ∈ I. Itfollows from Lemma 5.3 that Spu(T, S) = Sp(T,G) = E.


Let f ∈ L1(G) and ε > 0. There exists s ∈ S such that∫G\(S−s) |f(t)|dt <

ε. We set

fs(t) ={f(t− s) if t ∈ S

0 otherwise.

We have fs ∈ L1(S), ‖fs‖C(E) ≤ ‖f‖C(E) + ε and

fs(T ) =∫Sf(t− s)T (t)dt = T (s)

∫S−s

f(t)T (t)dt.

Since for every t, T (t) is an isometry with Spu(T, S) = E, we get∥∥∥∥∫S−s

f(t)T (t)dt∥∥∥∥ = ‖fs(T )‖ = lim

S‖T (t)fs(T )‖ ≤ c‖fs‖C(E) ≤ c(‖f‖C(E)+ε).

Hence

‖f(T )‖ ≤∥∥∥∥∫

S−sf(t)T (t)dt

∥∥∥∥+

∥∥∥∥∥∫G\(S−s)

f(t)T (t)dt

∥∥∥∥∥≤ c(‖f‖C(E) + ε) +

∫G\(S−s)

|f(t)|dt

≤ c‖f‖C(E) + (c+ 1)ε.

Letting ε → 0, we obtain ‖f(T )‖ ≤ c‖f‖C(E). Since f(T ) is the operatorπ(g)→ π(f ∗ g), we have ‖f(T )‖ = ‖π(f)‖ and thus

‖π(f)‖ ≤ c‖f‖C(E) (5.1)

We take I = J(E). It follows from (5.1) that if f ∈ L1(G) with f = 0 onE, then π(f) = 0 and thus f ∈ J(E). This shows that E satisfies spectralsynthesis.

We take now I = I(E). Let i : A(E)→ C0(E) be the canonical injection.Inequality (5.1) shows that the range of i is closed. On the other hand,since A(Γ) is dense in C0(Γ) ([21], Theorem 1.2.4, p. 9), the range of i isalso dense in C0(E). So i is surjective, which implies that E is a Helson setwith α(E) = ‖i−1‖ ≤ c. �

We get immediately the following result.

Corollary 5.5. Let E be a closed subset of Γ. The following are equivalents.(i) E satisfies spectral synthesis and is a Helson set with α(E) = 1.(ii) For every representation T of S by contractions on a Banach space

such that Spu(T, S) = E, we have

limS‖T (t)f(T )‖ = ‖f‖C(E), f ∈ L1(S).

In particular if T is a representation of S by contractions and if Spu(T, S) ={χ}, then

limS‖T (t)f(T )‖ = |f(χ)|, f ∈ L1(S).

16 M. ZARRABI

Remarks. In this remark we give some examples of Helson sets whichsatisfies spectral synthesis and discuss the constant α(E). Let E be a closedsubset of Γ; E is called independent if for every χ1, . . . , χk in E and everyintegers n1, . . . , nk,

χn11 . . . χnk1 = 1⇒ χn1

1 = . . . = χnkk = 1.

A compact subset E of Γ is called a Kronecker set if the following propertyholds: for every continuous function h on E with |h| = 1 and for every ε > 0there exists t ∈ G such that supχ∈E |h(χ)− χ(t)| < ε. Every Kronecker setE satisfies spectral synthesis and is a Helson set with α(E) = 1.

Indeed, every measure µ with support contained in a Kronecker set sat-isfies ‖µ‖ = ‖µ‖∞ (see [21], Theorem 5.5.2, p. 113). It follows from thediscussion made at the beginning of this section that a Kronecker set E is aHelson set with α(E) = 1.

The fact that a Kronecker set is of spectral synthesis was proved by N.Varopoulos ([26]) for totally disconnected Kronecker sets and then extendedto any Kronecker set by S. Saeki ([22]).

Notice that every Kronecker set is independent. If E is a finite indepen-dent set and if every element of E is of infinite order, then E is a Kroneckerset (see [21], p. 98).

We recall also that if E is a finite set, then E satisfies spectral synthesisand is a Helson set with α(E) ≤

√#E where #E is the cardinal of E.

On the other hand, for n a nonnegative integer, there exists E such that#E = 2n and α(E) ≥

√#E/2 (see [13], p. 33).

Finally we note that a countable, compact, independent set is a Helson setand satisfies spectral synthesis (see [21], Theorem 5.6.7, p. 117 and p.161).

6. Estimate of the limit with Holder space norm

The classical Wiener algebra A(T) is defined by

A(T) =

{h ∈ C(T) : ‖h‖1 =

+∞∑n=−∞

|h(n)| < +∞

},

where h(n) is the nth Fourier coefficient of h. We set

A+(T) ={h ∈ A(T) : h(n) = 0, n < 0

}.

A(T) may be regarded as the set of functions h = f , where f ∈ L1(Z) =`1(Z) and A+(T) as the set of functions h = f , with f ∈ `1(Z+).

Let E be a closed subset of T. It follows from Theorem 5.4 that theexistence of a constant c such that, limn→+∞ ‖Tn1 h(T1)‖ ≤ c‖h‖C(E), forevery h ∈ A+(T) and every contraction T1 with σ(T1) ∩ T = E, holds only


when E is a “thin” set. Our hope in this section is to investigate thisproperty for a large class of sets, but with a norm slightly bigger than thesupremum norm.

Let ϕ : R+ 7→ R+ be a nondecreasing function, with ϕ(0+) = 0. We willalways assume that the function t 7→ ϕ(t)/t is nonincreasing on (0,+∞).We denote by Λϕ(E) the set of all continuous functions h on E such that

∆ϕ,E(h) := sup{|h(z)− h(ζ)|ϕ(|z − ζ|)

, z, ζ ∈ E and z 6= ζ

}< +∞.

We set‖h‖Λϕ(E) = ‖h‖C(E) + ∆ϕ,E(h),

This defines a norm on Λϕ(E). For t > 0, we set

Et = {eis ∈ T, ∃ eiu ∈ E such that |s− u| ≤ t},

and we write |Et| for the Lebesgue measure of Et. Before stating our results,we first recall the following fact which will be useful. Assume thatϕ(t)|Et \ E|1/2 = o(t1/2), t → 0. If g ∈ A(T) ∩ Λϕ(T) vanishes on E, theng satisfies spectral synthesis for E. The proof is the same as in the caseϕ(t) = tγ , γ > 0 (see [15], p. 122-123).

Theorem 6.1. If ∫ 1

0ϕ(t)|Et|1/2t−3/2dt < +∞, (6.1)

then there exists a constant (depending only on E and ϕ) such that

limn→+∞

‖Tn1 h(T1)‖ ≤ c‖h‖Λϕ(E),

for every h ∈ A+(T) ∩ Λ+ϕ (T) and every contraction T1 on a Banach space

with σ(T1) ∩ T = E.

Proof. Let h ∈ A+(T)∩Λϕ(T) and T1 be a contraction on a Banach space X,with σ(T1) ∩ T = E. Let E = {t ∈ R, eit ∈ E} and let g be the 2π-periodiccontinuous function which is affine on the intervalles contiguous to E andcoincide on E with the function t→ h(eit). Since g is 2π-periodic, we shallconsider g as a function on T. It is shown in [15], Lemme I, p. 131, thatg ∈ Λϕ(T) and ωg(t) ≤ c0ωE,h(t), t > 0, where

ωg(t) = sup{|g(x)− g(x′)| : x, x′ ∈ R, |x− x′| ≤ t},

ωh,E(t) = sup{∣∣∣h (eix)− h(eix′)∣∣∣ : x, x′ ∈ E, |x− x′| ≤ t

},

and c0 is an absolute contant.Let

∑n∈Z γne

inx be the Fourier series of g. The proof given in [15], p. 132-

133, shows, with a minor modifications, that∑|n|>0 |γn| ≤ c1 supt>0

ωh,E(t)ϕ(t) ,

for some constant c1 depending only on E and ϕ. It follows that∑|n|>0 |γn| ≤

18 M. ZARRABI

c1∆ϕ,E(h). Moreover we have |γ0| ≤ supt∈R |g(t)| ≤ ‖h‖C(E). So g ∈ A(T)and

‖g‖1 ≤ c‖h‖Λϕ(E), (6.2)

where c = sup{1, c1}.If E = T we have h = g and then by (6.2), limn→+∞ ‖Tn1 h(T1)‖ ≤ ‖h‖1 ≤

c‖ϕ‖Λϕ(E).Assume now that E 6= T. Let M1 be the contraction defined on L(X) by

R 7→ T1R. It follows from Lemma 2.3 that there exists an isometry V suchthat σ(V ) ⊂ σ(M1) = σ(T1) and

limn→+∞

‖Tn1 h(T1)‖ = limn→+∞

‖Mn1 h(M1)I‖ ≤ ‖h(V )‖. (6.3)

We have σ(V )∩T ⊂ σ(T1)∩T. Hence σ(V ) 6= D and since V is an isometry,σ(V ) ⊂ T. Thus V is invertible. The function h−g belongs to A(T)∩Λϕ(T)and vanishes on E. Since ϕ is nondecreasing, condition (6.1) implies

ϕ(t)|Et|1/2∫ 2t

ts−3/2ds ≤

∫ 2t

tϕ(s)|Es|1/2s−3/2ds = o(1), as t→ 0.

Thus ϕ(t)|Et|1/2 = o(t1/2), t → 0. By the observation made just beforeTheorem 6.1, h−g satisfies spectral synthesis for E. Therefore h(V ) = g(V )(see [18], p. 201). It follows then from (6.3) and (6.2) that

limn→+∞

‖Tn1 h(T1)‖ ≤ ‖g(V )‖ ≤ ‖g‖1 ≤ c‖ϕ‖Λϕ(E).

�

The motivation for our hypothesis (6.1) comes from a known result whichsays that under condition (6.1) we have Λϕ(E) ⊂ A(E), where A(E) ={h|E , h ∈ A(T)} ([15], Theoreme IV, p. 133). Notice that for some sets,condition (6.1) is necessary and sufficient to have Λϕ(E) ⊂ A(E). We usethis fact to show that the same phenomenon holds for our results.

Let ξ ∈ (0, 1/2) and let Eξ be the perfect symetric set of ratio ξ,

Eξ ={e2iπ(1−ξ)

Pn≥1 εnξ

n−1

, εn = 0 or 1}.

For ξ = 1/3, Eξ is the classical triadic Cantor set. Notice that |Eξt | � t1−β,where β = − log 2

log ξ . It is shown in ([15], Theoreme VIII, p. 134), that theinclusion Λϕ(Eξ) ⊂ A(Eξ) holds if and only if condition (6.1) holds forE = Eξ.

We setΛ+ϕ (T) =

{h ∈ Λϕ(T) : h(n) = 0, n < 0

}.

A closed subset E ⊂ T is called an interpolating set for Λ+ϕ (T) if for every

function h ∈ Λϕ(E), there exists a function g ∈ Λ+ϕ (T) such that h = g on

E. Let γ ∈ (0, 1) and let ϕγ(t) = tγ , t > 0. Dyn’kin showed in [8] that


E is an interpolating set for Λ+ϕγ (T) if and only if E satisfies the following

condition : there exists a constant c > 0 such that

supz∈L

d(z, E) ≥ c|L|, for every arc L ⊂ T, (K)

where |L| is the length of L. Shirokov showed in [23] that the condition(K) also characterizes the interpolating sets for Λ+

ϕ (T), when ϕ satisfies thecondition ∫ t

0

ϕ(t)tdt ≤ const. ϕ(t), t > 0. (6.4)

Notice that any Eξ satisfies condition (K) and thus is an interpolating setfor Λ+

ϕ (T), when ϕ satisfies (6.4). We have the following result which showsthat condition (6.1) is sharp for the sets Eξ.

Theorem 6.2. Assume that ϕ satisfies condition (6.4). Then the followingare equivalent.

(i)∫ 1

0 ϕ(t)|Eξt |1/2t−3/2dt < +∞.

(ii)∫ 1

0 ϕ(t)t−1−β2 dt < +∞, where β = − log 2

log ξ .(iii) There exists a constant c such that

limn→+∞

‖Tn1 h(T1)‖ ≤ c‖h‖Λϕ(Eξ),

for every h ∈ A+(T) ∩ Λ+ϕ (T) and every contraction T1 on a Banach space,

with σ(T1) ∩ T = Eξ.

For the proof of this theorem, we need the following lemma.

Lemma 6.3. Assume that ϕ satisfies condition (6.4) and

∫ 1

0ϕ(t)t−1−β

2 dt = +∞, (6.5)

Then there exists a nondecreasing function ψ : (0,+∞) 7→ (0,+∞) suchthat

(a) t 7→ ψ(t)t is nonincreasing.

(b) ψ(t) = o(ϕ(t)), as t→ 0.(c)

∫ t0ψ(t)t dt ≤ const. ψ(t), t > 0.

(d)∫ 1

0 ψ(t)t−1−β2 dt = +∞.

Proof. One may construct, by induction, a decreasing sequence (tn)n≥0 in(0, 1) tending to 0, such that

tn+1 <n+ 1n+ 2

tn and∫ tn

tn+1

ψ(t)t−1−β2 dt ≥ 1. (6.6)

20 M. ZARRABI

Let ν : (0,+∞)→ (0,+∞) be the continuous function defined by

ν(t) =1

n+ 1if t = tn, ν is affine on [tn+1, tn], and ν(t) = 1 for t ≥ t0.

Clearly ν is increasing and limt→0 ν(t) = 0. Moreover the function t→ ν(t)t

is decreasing since by (6.6) ν(tn+1)tn+1

> ν(tn)tn

and on each interval [tn+1, tn],

ν(t)t

=1

(n+ 1)(n+ 2)(tn − tn+1)

(1 +

(n+ 1)tn − (n+ 2)tn+1

t

),

is decreasing.Since t 7→ ϕ(t)

t is nonincreasing, we obtain∫ 1

sϕ(t)t−1−β

2 dt ≤ ϕ(s)s

∫ 1

st−

β2 dt ≤ 2

2− βϕ(s)s, 0 < s < 1,

and since ϕ satisfies condition (6.5), we get lims→0ϕ(s)s = +∞. In particular

we have limt→0 ν(

tϕ(t)

)= 0.

We set ψ(t) = ν(

tϕ(t)

)ϕ(t), t > 0. Then ψ is nondecreasing and we

have ψ(t) = o (ϕ(t)), t → 0. Since the functions t 7→ ν(t)t and t 7→ ϕ(t)

t arenonincreasing, it is easily seen that the function ψ(t)

t is nonincreasing too.On the other hand we have∫ t

0

ψ(s)s

ds ≤ ν(

t

ϕ(t)

)∫ t

0

ϕ(s)sds

≤ const. ν(

t

ϕ(t)

)ϕ(t) = const. ψ(t), t > 0.

It remains now to show property (d). Let n0 be such that ϕ(t) ≤ 1 fort ≤ tn0 . Thus ν

(t

ϕ(t)

)≥ ν(t) ≥ ν(tn+1) for t ∈ [tn+1, tn] and n ≥ n0. We

have ∫ tn0

0ψ(t)t−1−β

2 dt =∑n≥n0

∫ tn

tn+1

ν

(t

ϕ(t)

)ϕ(t)t−1−β

2 dt

≥∑n≥n0

ν(tn+1)∫ tn

tn+1

ϕ(t)t−1−β2 dt

≥∑n≥n0

1n+ 2

= +∞.

�

Proof of Theorem 6.2. The equivalence between (i) and (ii) follows fromthe fact that |Eξt | � t1−β. The implication (i)⇒ (iii) follows from Theorem6.1.


We shall now show (iii)⇒ (ii). We consider the Banach space

A(Eξ) ={h|Eξ , h ∈ A(T)

},

endowed with the quotient norm of A(T)/I, where I = {h ∈ A(T), h =0 on E}. Let T : A(Eξ) 7→ A(Eξ) be the contraction defined by h 7→ (z 7→zh(z)); T is an isometry with σ(T ) = Eξ. Since we assume that (iii) holds,we have for every h ∈ A+(T) ∩ Λ+

ϕ (T),

‖h|Eξ‖A(Eξ) = ‖h(T )‖ = limn→+∞

‖Tnh(T )‖ ≤ c‖h‖Λϕ(Eξ).

It follows that F ‖.‖Λϕ(Eξ) ⊂ A(Eξ), where F ={h|Eξ , h ∈ A+(T) ∩ Λ+

ϕ (T)}

.

But we have F ‖.‖Λϕ(Eξ) = λ+ϕ (Eξ) := {h|Eξ , h ∈ λ+(T)}, where

λ+ϕ (T)} =

{h ∈ Λ+

ϕ (T),|h(z)− h(ξ)|ϕ(|z − ξ|)

→ 0, as |z − ξ| → 0, z 6= ξ

}.

Let ψ : (0,+∞) 7→ (0,+∞) be any nondecreasing function satisfyingproperties (a), (b) and (c) of Lemma 6.3. By the property (b), we haveΛ+ψ (Eξ) ⊂ λ+

ϕ (Eξ). Thus Λ+ψ (Eξ) ⊂ A(Eξ). Since ψ satisfies (c) and since

Eξ satisfies condition (K), Eξ is an interpolating set for Λ+ψ (T), that is

Λ+ψ (Eξ) = Λψ(Eξ) ([23]). Therefore we obtain Λψ(Eξ) ⊂ A(Eξ). Since ψ

is nondecreasing and satisfies (a), it follows from ([15], Theoreme VIII, p.134), that ∫ 1

0ψ(t)t−1−β

2 dt < +∞.

By Lemma 6.3 (d), we deduce that (ii) of Theorem 6.2 holds. �

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Universite de Bordeaux, UMR 5251, 351, cours de la Liberation, F-33405

Talence cedex, France

E-mail address: [email protected]

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