On the growth of powers of operators with spectrum contained in Cantor sets

0.13 MB·English

by  Cyril Agrafeuil

On the growth of powers of operators with spectrum contained
in Cantor sets
6
AGRAFEUIL Cyril
0
0
2

n Abstract
a
J For ξ ∈ 0,12 , we denote by Eξ the perfect symmetric set associated to ξ, that is
(cid:0) (cid:1)
5
+∞
2
E = exp 2iπ(1−ξ) ǫ ξn−1 : ǫ =0 or 1 (n≥1) .
ξ n n
] n (cid:0) nX=1 (cid:1) o
A
Letsbeanonnegativerealnumber,andT beaninvertibleboundedoperatoronaBanach
F
space with spectrum included in E . We show that if
. ξ
h
t Tn =O ns , n→+∞
a
m (cid:13)(cid:13) (cid:13)(cid:13) (cid:0) (cid:1) log1 −log2
and T−n =O enβ , n→+∞ for some β < ξ ,
[ (cid:13)(cid:13) (cid:13)(cid:13) (cid:0) (cid:1) 2logξ1 −log2
1
v then for every ε>0, T satisfies the stronger property
9
0 T−n =O ns+12+ε , n→+∞.
6 (cid:13) (cid:13) (cid:0) (cid:1)
1 This result is a particular ca(cid:13)se of(cid:13)a more general result concerning operators with spec-
0 trum satisfying some geometrical conditions.
6
0
h/ 1 Introduction
t
a
m We denote by T the unit circle and by D the open unit disk. We shall say that a closed
: subset E of T is a K-set if there exists a positive constant c such that for any arc L of T,
v
i
X supd(z,E) ≥ c|L|, (K)
r z L
a ∈
where |L| denotes the length of the arc L and d(z,E) the distance between z and E. Let E
be a K-set. We set
2π 1
δ(E) = sup δ ≥ 0 : dt< +∞ .
n Z0 d(eit,E)δ o
log 1
We have δ(E) ≥ 1−c (see [2] section 5, proof of lemma 2 and corollary). E. M. Dyn’kin
log 2
1 c
showed in [2] that con−dition (K) characterizes the interpolating sets for Λ+(T), s > 0 (see
s
1
section 2 for the definition of Λ+(T)). Let s be a nonnegative real number, and let T be an
s
invertible operator on a Banach space. We show (theorem 2.3) that if the spectrum of T is
included in E and if T satisfies
Tn = O ns , n → +∞
(cid:13) (cid:13) (cid:0) (cid:1)
and (cid:13)T (cid:13)n = O enβ , n→ +∞ for some β < δ(E) ,
−
1+δ(E)
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
then for every ε > 0, T also satisfies the stronger property
T−n = O ns+21+ε , n → +∞. (1)
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
1
For ξ ∈ 0, , we denote by E the perfect symmetric set associated to ξ, that is
ξ
(cid:16) 2(cid:17)
+
∞
E = exp 2iπ(1−ξ) ǫ ξn 1 : ǫ = 0 or 1 (n ≥ 1) .
ξ n − n
n (cid:0) nX=1 (cid:1) o
log 1 −log2
ξ
We set b(ξ) = . We obtain (as a consequence of theorem 2.3) that if the spec-
2log 1 −log2
ξ
trum of T is included in E , Tn = O ns , n → +∞ and T n = O enβ , n → +∞
ξ −
for some β < b(ξ), then T sat(cid:13)isfies(cid:13)(1). N(cid:0)ot(cid:1)ice that J. Esterle(cid:13)show(cid:13)ed in [(cid:0)5] th(cid:1)at if T is a
(cid:13) (cid:13) (cid:13) (cid:13)
contraction on a Banach space (respectively on a Hilbert space) with spectrum included in
1
E (respectively included in E ) such that T n = O enβ , n → +∞ for some β < b
1 ξ −
q (cid:13) (cid:13) (cid:0) (cid:1) (cid:0)q(cid:1)
(respectively β < b(ξ)), then sup T n <+(cid:13)∞ (re(cid:13)spectively T is an isometry). Here q is an
−
n 0(cid:13) (cid:13)
integer greater than or equal t≥o 3(cid:13). (cid:13)
2 Growth of powers of operators
Let p be a non-negative integer. We denote by Cp(T) the space of p times continuously
differentiable functions on T. We set
a
p(D) = f ∈ Cp(T): f(n) = 0 (n < 0) ,
n o
a a b
C (T)= Cp(T) and (D)= p(D).
∞ p 0 ∞ p 0
Let s be Ta n≥onnegative real numbeTr, ≥we denote by [s] the nonnegative integer such that
[s]≤ s < [s]+1. We define the Banach algebra
f([s])(z)−f([s])(z )
Λ (T) = f ∈ C[s](T) : sup ′ < +∞ ,
s n z,z′ T (cid:12)(cid:12) |z−z′|s−[s] (cid:12)(cid:12) o
∈
2
f([s])(z)−f([s])(z )
′
equiped with the norm f = f + sup . We also define the
(cid:13) (cid:13)Λs (cid:13) (cid:13)C[s](T) z,z′ T(cid:12)(cid:12) |z−z′|s−[s] (cid:12)(cid:12)
subalgebra (cid:13) (cid:13) (cid:13) (cid:13) ∈
λ (T)= f ∈ C[s](T) : f([s])(z)−f([s])(z ) = o |z−z |s [s] , |z−z | → 0 ,
s ′ ′ − ′
n (cid:12) (cid:12) (cid:0) (cid:1) o
(cid:12) (cid:12)
which we equip with the same norm. We also set
Λ+(T) = f ∈ Λ (T) : f(n)= 0 (n < 0)
s s
n o
and λ+(T) = f ∈ λ (T) : fb(n) = 0 (n < 0) .
s s
n o
b a
We remark that if s is an integer, Λ (T) = λ (T) = Cs(T) and so Λ+(T) = λ+(T) = s(D).
s s s s
We define
N (E) = f ∈ Λ (T) : f = ... = f([s]) = 0 ,
s s
(cid:8) |E |E (cid:9)
and set N+(E) = N (E)∩Λ+(T).
s s s
Lemma 2.1. Let s be a nonnegative real number. Then for all ε > 0, we have the following
continuous embedding
Λ (T) ֒→ A (T).
s+1+ε s
2
Proof. For s = 0, this is a result of Bernstein (see [9], p.13). The general case is obtained by
the same arguments. Let ε > 0, and set s˜= s+ 1 +ε. Let f ∈Λ (T). For h > 0, define
2 s˜
2π
P(h) = f([s˜])(ei(t h))−f([s˜])(ei(t+h)) 2dt.
−
Z
0 (cid:12) (cid:12)
(cid:12) (cid:12)
It follows from Parseval equality that
+
P(h) = 8π ∞ f[([s˜])(n) 2sin2(nh). (2)
n=X (cid:12) (cid:12)
−∞(cid:12) (cid:12)
Let j be the smallest integer such that [s˜] < 2j0 and let j ≥ j . It follows from the relation
0 0
[ [s˜]
f([s˜])(n) = (n+k) f(n+[s˜])(n ∈ Z) and from (2) that there exists a constant C > 0
1
(cid:16)kQ=1 (cid:17)
independent of f such thbat
2j+1 1
P(h) ≥ 4 − f(n+[s˜]) 2(1+|n|)2[s˜]sin2(nh). (3)
C2
1 nX=2j (cid:12)(cid:12)b (cid:12)(cid:12)
| |
Using the Cauchy-Schwartz inequality, we have
2j+1 1 2j+1 1 1 2j+1 1 1
− f(n+[s˜]) (1+|n|)s ≤ − f(n+[s˜]) 2(1+|n|)2[s˜] 2 − (1+|n|)2s 2[s˜] 2 (4)
−
nX=2j (cid:12)(cid:12)b (cid:12)(cid:12) (cid:16) nX=2j (cid:12)(cid:12)b (cid:12)(cid:12) (cid:17) (cid:16) nX=2j (cid:17)
| | | |
3
π π 2π
Set h = . For all integers n such that 2j ≤ |n| ≤ 2j+1−1, we have ≤ |nh| ≤ , and
3.2j 3 3
1
so sin2(nh)≥ . So, we deduce from (3) that
4
2j+1−1 f(n+[s˜]) 2(1+|n|)2[s˜] 21 ≤ C P π 21.
(cid:16) nX=2j (cid:12)(cid:12)b (cid:12)(cid:12) (cid:17) 1 (cid:0)3.2j(cid:1)
| |
Then, as f ∈ Λ (T), we have
s˜
π 1 1 2π s˜ [s˜]
P 2 ≤ (2π)2 f − ,
(cid:0)3.2j(cid:1) (cid:13) (cid:13)Λs˜(cid:0)3.2j(cid:1)
(cid:13) (cid:13)
so that
(cid:16)2njX+=1−2j1(cid:12)(cid:12)fb(n+[s˜])(cid:12)(cid:12)2(1+|n|)2[s˜](cid:17)21 ≤ C1(2π)12(cid:13)(cid:13)f(cid:13)(cid:13)Λs˜(cid:0)32.2πj(cid:1)s˜−[s˜]. (5)
| |
Furthermore, there exists a constant C > 0 such that
2
2j+1 1 1
− (1+|n|)2s−2[s˜] 2 ≤ C22j s−[s˜]+21 . (6)
(cid:16) X (cid:17) (cid:0) (cid:1)
n=2j
| |
Finally we deduce from (4) and the inequalities (5) and (6) that there exists a constant
C > 0 independent of f such that for all j ≥ j ,
3 0
2j+1 1
X− (cid:12)f(n+[s˜])(cid:12)(1+|n|)s ≤ 2j(s−s˜+21)C3(cid:13)f(cid:13)Λs˜ = 2−εjC3(cid:13)f(cid:13)Λs˜.
n=2j (cid:12)b (cid:12) (cid:13) (cid:13) (cid:13) (cid:13)
| |
Summing over j ≥ j these inequalities, we get
0
C
f(n+[s˜]) (1+|n|)s ≤ 3 f ,
nX2j0(cid:12)(cid:12)b (cid:12)(cid:12) 1−2−ε(cid:13)(cid:13) (cid:13)(cid:13)Λs˜
| |≥
On the other hand, we have f(n) ≤ f for every n ∈ Z. So, since j is independent of
f, there exists a constant K >(cid:12) 0 (i(cid:12)ndep(cid:13)en(cid:13)dΛes˜nt of f) such that 0
(cid:12)b (cid:12) (cid:13) (cid:13)
f ≤ K f
(cid:13) (cid:13)s (cid:13) (cid:13)Λs˜
(cid:13) (cid:13) (cid:13) (cid:13)
Before giving the main theorem of the paper, we need the following lemma.
4
Lemma 2.2. Let E be a closed subset of T. We assume that there exists δ > 0 for which
2π 1 δ
dt < +∞. Let β < and let T be an invertible operator on a Banach
Z d(eit,E)δ 1+δ
0
space with spectrum included in E that satisfies
Tn = O ns , n → +∞ ( for some nonnegative real s)
and (cid:13)(cid:13)T (cid:13)(cid:13)n = O(cid:0) e(cid:1)nβ , n→ +∞,
−
(cid:13) (cid:13) (cid:0) (cid:1) a
Then there exists an (cid:13)outer(cid:13)function f ∈ (D) which vanishes exactly on E and such that
∞
+
f(T):= ∞f(n)Tn = 0.
nP=0
b
Proof. Let ω be the weight defined by ω(n) = Tn (n ∈ Z). Let Φ be the continuous
morphism from A (T) to L(X) defined by (cid:13) (cid:13)
ω (cid:13) (cid:13)
+
∞
Φ(f)= f(T)= f(n)Tn f ∈A (T) .
ω
n=X (cid:0) (cid:1)
−∞ b
Since the algebra A (T) is regular, we have z ∈ T : f(z) = 0 (f ∈ Ker Φ) ⊂ E (see [7],
ω
theorem 2.5), and so J (E) ⊂ Ker Φ. Then(cid:8)the result follows from lemmas(cid:9)7.1 and 7.2 of
ω
[5].
Theorem2.3. LetE beaK-set, andletsbeanonnegative realnumber. Then, anyinvertible
operator T on a Banach space with spectrum included in E that satisfies
Tn = O ns , n → +∞
(cid:13) (cid:13) (cid:0) (cid:1)
and (cid:13)T (cid:13)n =O enβ , n → +∞ for some β < δ(E) ,
−
1+δ(E)
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
also satisfies the stronger property
T−n = O ns+21+ε , n → +∞,
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
for all ε > 0.
Proof. Let ε > 0 and set s˜ = s+ 1 +ε. Without loss of generality, we may assume that s˜
2
is not an integer. Let t a real number, which is not an integer, and satisfies s+ 1 < t < s˜
2
and [t] = [s˜]. According to lemma 2.1, we can define a continuous morphism Φ from λ+(T)
t
to L(X) by
+
∞
Φ(f)= f(T)= f(n)Tn f ∈ λ+(T) .
t
nX=0 (cid:0) (cid:1)
b
Let I = Ker Φ, I is a closed ideal of λ+(T). We denote by S its inner factor, that is the
t I
greatest common divisor of all inner factors of the non-zero functions in I (see [8] p.85), and
5
we set, for 0 ≤k ≤ [t], hk(I) = z ∈ T : f(z)= ... = f(k)(z) = 0 (f ∈ I) .
F. A. Shamoyan showed in [12](cid:8)that (cid:9)
I = f ∈ λ+(T): S(I)|S(f) and f(k) = 0 on hk(I) for all 0≤ k ≤ [t] ,
t
n o
where S(f) denotes the inner factor of f and S(I)|S(f) means that S(f)/S(I) is a bounded
δ(E)
holomorphic function in D. Since β < , there exists 0 < δ < δ(E) such that
1+δ(E)
δ 2π 1
β < . We have, by definition of δ(E), dt < +∞. So we deduce from
1+δ Z d(eit,E)δ
0 a
lemma 2.2 that there exists an outer function f ∈ (D) which vanishes exactly on E and
∞
such that f ∈ I. Therefore, we have S(I) = 1 and h0(I) ⊂ E, so that N+(E)∩λ (T) ⊂ I.
t t
Now, as Λ+(T) ⊂ λ+(T), we can define a continuous morphism Ψ from Λ+(T) to L(X) by
s˜ t s˜
Ψ = Φ . Using what precedes, we have
|Λs+˜(T)
N+(E) ⊂ Ker Ψ.
s˜
SothereexistsacontinuousmorphismΨ˜ fromΛ+(T)/N+(E)intoL(X)suchthatΨ = Ψ˜◦π+,
s˜ s˜ s˜
where π+ is the canonical surjection from Λ+(T) to Λ+(T)/N+(E). Since E is a K-set, by
s˜ s˜ s˜ s˜
a theorem of E. M. Dyn’kin [2], it is an interpolating set for Λ+(T), so that the canonical
s˜
imbedding i from Λ+(T)/N+(E) into Λ (T)/N (E) is onto. We have, for n ≥ 0,
s˜ s˜ s˜ s˜
T n = Ψ˜ ◦i 1◦π (α n),
− − s˜ −
where π denote the canonical surjection from Λ (T) to Λ (T)/N (E) and where α : z → z
s˜ s˜ s˜ s˜
is the identity map. So we have, for n ≥ 0,
T n ≤ Ψ˜ ◦i 1 π (α n)
(cid:13) − (cid:13) (cid:13) − (cid:13)(cid:13) s˜ − (cid:13)Λs˜
(cid:13) (cid:13) ≤ (cid:13)Ψ˜ ◦i 1(cid:13)((cid:13)1+n)s˜, (cid:13)
−
(cid:13) (cid:13)
(cid:13) (cid:13)
which completes the proof.
We give two immediate corollaries of this theorem.
Corollary 2.4. Let ξ ∈ 0,1 and let s be a nonnegative real number. Then, any invertible
2
operator T on a Banach(cid:0)space(cid:1)with spectrum included in E that satisfies
ξ
Tn =O ns , n→ +∞
and (cid:13)(cid:13)T (cid:13)(cid:13)n = O(cid:0) e(cid:1)nβ , n → +∞ for some β < b(ξ),
−
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
also satisfies the stronger property
T−n = O ns+21+ε , n → +∞,
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
for all ε > 0.
6
Proof. It is well known that E is a K-set (see proposition 2.5 of [5]). Moreover, E satisfies
ξ ξ
2π 1 log2 2π 1
dt< +∞ifandonly ifδ < 1+ . Indeed,thecondition dt <
Z d(eit,E)δ logξ Z d(eit,E)δ
0 0
+ 2n−1
+∞ is equivalent to ∞ Ln,i 1−δ, where Ln,i are the arcs contiguous to Eξ, and Ln,i
nP=1 iP=1 (cid:12) (cid:12) (cid:12) (cid:12)
are their length, which is equ(cid:12)al to(cid:12) 2πξn 1(1−2ξ) (see [10] for further details). Then(cid:12)it is(cid:12)
−
log2 log2
easily seen that the last series converges if and only if δ < 1+ , so δ(E ) = 1+ .
ξ
logξ logξ
Now, the result follows immediately from theorem 2.3.
Then we obtain an other immediate result, which generalizes theorem 4.1 of [3]. Indeed,
the condition ” T n = O enβ , n → +∞” which appears in the following corollary is
−
weaker than the(cid:13)cond(cid:13)ition us(cid:0)ed b(cid:1)y the authors of [3].
(cid:13) (cid:13)
Corollary 2.5. Let E be a K-set, and let s be a nonnegative real number. Then, there exists
a constant β > 0 independent of s such that any invertible operator T on a Banach space
with spectrum included in E that satisfies
Tn = O ns , n → +∞
and (cid:13)(cid:13)T (cid:13)(cid:13)n = O(cid:0) e(cid:1)nβ , n → +∞,
−
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
also satisfies the stronger property
T−n = O ns+21+ε , n → +∞,
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
for all ε > 0.
Proof. As E is a K-set, we deduce from [2] (section 5, corollary) that δ(E) > 0. Then the
δ(E)
result follows immediately from theorem 2.3, with any β < .
1+δ(E)
Remark 2.6:
1) Someresults concerning operators with countable spectrumare obtained in [13] andin [1].
Let E be a closed subset of T and let s, t be two nonnegative reals. We denote by P(s,t,E)
the following property: every invertible operator T on a Banach space such that SpT ⊂ E
and satisfies the conditions:
Tn =O(ns) (n → +∞)
(cid:13) (cid:13)
(cid:13)T (cid:13)n = O(eε√n) (n → +∞), for all ε> 0,
−
(cid:13) (cid:13)
(cid:13) (cid:13)
also satisfies the stronger property
T n = O(nt)(n → +∞).
−
(cid:13) (cid:13)
M. Zarrabi showed in [13] (th(cid:13)´eor`em(cid:13)e 3.1 and remarque 2.a) that a closed subset E of T
satisfies P(0,0,E) if and only if E is countable. Notice that E is called a Carleson set if
2π 1
log+ dt < +∞. If E is a countable closed subset of T, we show in [1] that the
Z d(eit,E)
0
7
following conditions are equivalent:
(i) there exist two positive constants C ,C such that for every arc I ⊂ T,
1 2
1 1 1
log+ dt ≤ C log +C .
|I| Z d(eit,E) 1 |I| 2
I
(ii) E is a Carleson set and for all s ≥ 0, there exists t such that P(s,t,E) is satisfied.
For contractions with spectrum satisfying the Carleson condition, we can see [11].
1 1
2) When ξ = , the constant b in corollary 2.4 is the best possible in view of [6],
q q
(cid:0) (cid:1)
where the authors built a contraction T such that lim log T n = +∞, SpT ⊂ E and
− 1
log T−n = O nb(q1) . According to theorem 6.4 onf→[+4∞], T d(cid:13)(cid:13)oesn’(cid:13)(cid:13)t satisfy T−n = Oq ns
for(cid:13)any re(cid:13)al s≥(cid:0)0. (cid:1) (cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13) (cid:13) (cid:13)
References
[1] C.Agrafeuil,Id´eauxferm´esdecertainesalg`ebresde Beurlingetapplication auxop´erateurs
a` spectre d´enombrable, preprint.
[2] E. M. Dynkin, Free interpolation set for Ho¨lder classes, Mat. Sbornik, 109 (1979), 107-
128.
[3] O. El-Fallah et K. Kellay, Sous-espaces biinvariants pour certains shifts pond´er´es, Ann.
Inst. Fourier 48 (1998), no. 5, 1543-1558.
[4] J. Esterle, Uniqueness, strong form of uniqueness and negative powers of contractions,
Banach Center Publ. 30 (1994), 1-19.
[5] J. Esterle, Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals
of A+, J. reine angew. Math. 450 (1994), 43-82.
[6] J. Esterle, M. Rajoelina and M. Zarrabi, On contractions with spectrum contained in
Cantor set, Math. Proc. Camb. Phil. Soc. 117 (1995), 339-343.
[7] J. Esterle, E. Strouse and F. Zouakia, Theorems of Katznelson-Tzafriri type for contrac-
tions, J. Func. Anal. (2) 94 (1990), 273-287.
[8] K. Hoffman ”Banach spaces of analytic functions”, Prentic-Hall, Englewood Cliffs, 1962.
[9] J. P. Kahane, ”S´eries de Fourier absolument convergentes”, Erg. Math. 336, Springer
Verlag, Berlin-Heidelberg-New York, 1973.
[10] J. P. Kahane, R. Salem, ”Ensembles parfaits et s´eries trigonom´etriques”, Paris, Her-
mann, 1963.
[11] K. Kellay, Contractions et hyperdistributions a` spectre de Carleson, J. London Math.
Soc. (2) 58 (1998), 185-196.
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[12] F. A. Shamoyan Closed ideals in algebras of functions analytic in the disc and smooth
up to its boundary, Mat. Sbornik 79 (1994), 425-445.
[13] M. Zarrabi, Contractions a` spectre d´enombrable et propri´et´e d’unicit´e des ferm´es
d´enombrables du cercle unit´e, Ann. Inst. Fourier 43 (1993), 251-263.
2000 Mathematics Subject Classification: 46J15, 46J20, 47A30.
Key-words: operators, Beurling algebra, spectral synthesis, perfect symmetric set.
AGRAFEUIL Cyril
[e-mail]
Laboratoire Bordelais d’Analyse et G´eom´etrie (LaBAG), CNRS-UMR 5467 Universit´e Bor-
deaux I
351, cours de la lib´eration
33405 Talence cedex, FRANCE.
9