Classical operators on weighted Banach spaces of entire functions · Classical operators on...

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Introduction Classical results Weighted Banach spaces of holomorphic functions Continuity, norms and spectrum Dynamics of D and J on H aand H 0 aClassical operators on weighted Banach spaces of entire functions Mar´ ıa Jos´ e Beltr´ an Meneu Joint work with Jos´ e Bonet and Carmen Fern´ andez Congreso RSME 2013 Classical operators on weighted Banach spaces of entire functions 1/21

Transcript of Classical operators on weighted Banach spaces of entire functions · Classical operators on...

Page 1: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Classical operators on weighted Banachspaces of entire functions

Marıa Jose Beltran Meneu

Joint work with

Jose Bonet and Carmen Fernandez

Congreso RSME 2013

Classical operators on weighted Banach spaces of entire functions 1/21

Page 2: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Aim of the talkDynamics on operators

Aim of the talk

To study the dynamics of the operators:

Differentiation: Df := f ′

Integration: Jf (z) :=∫ z

0f (ξ)dξ, z ∈ C

Hardy operator: Hf (z) := 1z

∫ z

0f (ξ)dξ, z ∈ C

on weighted Banach spaces of entire functions.

D, J and H are continuous on (H(C), co), where co denotes thecompact-open topology.

DJf = f and JDf (z) = f (z)− f (0) ∀f ∈ H(C), z ∈ C.

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Page 3: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Aim of the talkDynamics on operators

Dynamics on operators

Given a Banach space X ,

L(X ) := {T : X → X linear and continuous }.

Given T ∈ L(X ), the pair (X ,T ) is a linear dynamical system.

Definitions

Let x ∈ X . The orbit of x under T is the set

Orb(x ,T ) := {x ,Tx ,T 2x , ...} = {T nx : n ≥ 0}.

x ∈ X is a periodic point if ∃n ∈ N such that T nx = x .

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Page 4: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Aim of the talkDynamics on operators

Dynamics on operators

Given a Banach space X and T ∈ L(X ), it is said that:

Definitions

T topologically mixing ⇔ ∀U,V 6= ∅ open, ∃n0 : T nU ∩ V 6= ∅∀n ≥ n0.

T hypercyclic ⇔ ∃x ∈ X , Orb(T , x) := {x ,Tx ,T 2x , . . . } is densein X ⇒ X SEPARABLE!!!

Definition (Godefroy, Shapiro, 1991)

T is chaotic if

T has a dense set of periodic points,

T is hypercyclic.

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Page 5: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Aim of the talkDynamics on operators

Dynamics on operators

Given a Banach space X and T ∈ L(X ), it is said that:

Definitions

T power bounded ⇔ supn ‖T n‖ <∞T Cesaro power bounded ⇔ supn ‖ 1n

∑nk=1 T

k‖ <∞T mean ergodic ⇔

∀x ∈ X , ∃Px := limn→∞

1

n

n∑k=1

T kx ∈ X

T uniformly mean ergodic ⇔{1

n

n∑k=1

T k

}n

converges in the operator norm.

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Page 6: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

D and J on H(C)

Classical results

Mac Lane (1952)

D : H(C)→ H(C) is hypercyclic, i.e.,

∃f0 ∈ H(C) : ∀f ∈ H(C), ∃(nk)k ⊆ N such that

f(nk )0 → f uniformly on compact sets.

Proposition

The integration operator J : H(C)→ H(C) is not hypercyclic.

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Page 7: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

The spaces Hυ (C) and H0υ (C)

υ : C→]0,∞[ is a weight if it is continuous, radial, i.e. υ(z) = υ(|z |),υ(r) is non-increasing on [0,∞[ and limr→∞ rmυ(r) = 0 ∀ m ∈ N.

Definition

Given a weight υ, the weighted Banach spaces of entire functions:

H∞υ := {f ∈ H(C) : ‖f ‖υ := supz∈C

υ(z)|f (z)| <∞}

H0υ := {f ∈ H(C) : lim

|z|→∞υ(z)|f (z)| = 0}.

Given a ∈ R, α > 0, consider υa,α(z) := |z |ae−α|z|, for |z | ≥ r0, and thespaces H∞a,α and H0

a,α. For a = 0, denote them by H∞α and H0α.

f ∈ H∞α ⇔ ∃C > 0 : |f (z)| ≤ Ceα|z| ∀z ∈ C.H∞α∼= `∞ and H0

α∼= c0 (Lusky).

P are dense in H0α but the monomials are not a Schauder basis.

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Page 8: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Differentiation operatorIntegration operatorHardy operator

Continuity, norms and spectrum

Lemma

Assume T : (H(C), co)→ (H(C), co) continuous and T (P) ⊆ P. TFAE:

(i) T (H∞υ ) ⊆ H∞υ ,

(ii) T : H∞υ → H∞υ is continuous,

(iii) T (H0υ) ⊆ H0

υ,

(iv) T : H0υ → H0

υ is continuous.

If this holds, ‖T‖L(H∞υ ) = ‖T‖L(H0

υ)and σH∞

υ(T ) = σH0

υ(T ),

where σX (T ) := {λ ∈ C : T − λI has no inverse }.

Harutyunyan, Lusky: The continuity of D and J on H∞υ (C) is determinedby the growth or decline of υ(r)eαr for some α > 0 in an interval [r0,∞[.

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Page 9: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Differentiation operatorIntegration operatorHardy operator

Continuity, norms and spectrum

If υ(r) = r ae−αr (α > 0, a ∈ R) for r ≥ r0 : ||zn||a,α ≈ ( n+aeα )n+a, with

equality for a = 0.

Proposition

For a > 0:

||Dn||a,α = O(n!(

eαn−a

)n−a)and n!

(eαn+a

)n+a

= O(||Dn||a,α)

For a ≤ 0 :

||Dn||a,α ≈ n!

(eα

n + a

)n+a

and the equality holds for a = 0.

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Page 10: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Differentiation operatorIntegration operatorHardy operator

Continuity, norms and spectrum

Proposition

For every α > 0 and a ∈ R, the spectrum σa,α(D) = αD.

Proposition

Let υ be a weight such that D is continuous on H∞υ (C) and that υ(r)eαr

is non increasing for some α > 0. If |λ| < α, the operator D − λI issurjective on H∞υ (C) and on H0

υ(C) and it even has a continuous linearright inverse

Kλf (z) := eλz∫ z

0

e−λξf (ξ)dξ, z ∈ C

In particular, this is satisfied by the weight υa,α(r) = r ae−αr for r bigenough (proved by Atzmon, Brive (2006), in the case a = 0).

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Page 11: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Differentiation operatorIntegration operatorHardy operator

Continuity, norms and spectrum

Proposition

For the weight υ(r) = r ae−αr (α > 0, a ∈ R) for r big enough, we have:

||Jn||a,α ∼= 1/αn, with the equality for a = 0,

σa,α(J) = (1/α)D,J − λI is not surjective on H∞a,α or H0

a,α if |λ| ≤ 1/α.

Classical operators on weighted Banach spaces of entire functions 11/21

Page 12: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

Differentiation operatorIntegration operatorHardy operator

Continuity, norms and spectrum

Theorem

For υ an arbitrary weight, the Hardy operator H : H∞υ (C)→ H∞υ (C) iscontinuous with norm ‖H‖υ = 1. Moreover, H2(H∞υ (C)) ⊂ H0

υ(C) andH2 is compact. Therefore, σ(H) = { 1n}N ∪ {0}. If the integrationoperator J : H∞υ (C)→ H∞υ (C) is continuous, then H is compact.

Remark

For the weight υ(r) = exp(−(log r)2) :

J is not continuous on H∞υ (C) (Harutyunyan, Lusky)

H : H∞υ (C)→ H0υ(C), H : H0

υ(C)→ H0υ(C), are compact (Lusky).

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Page 13: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

HypercyclicityMean ergodicity

Hypercyclicity

Theorem (Bonet, 2009)

D : H0a,α → H0

a,α satisfy:

0 < α < 1 =⇒ D is not hypercyclic and has no periodic pointdifferent from 0.

α = 1 =⇒ if a < 1/2, then D is topologically mixing, and ifa ≥ 1/2, D is not hypercyclic. It has no periodic point different from0 iif a ≥ 0.

α > 1 =⇒ D is chaotic and topologically mixing.

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Page 14: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

HypercyclicityMean ergodicity

Mean ergodicity

Remark

T ∈ L(X ) Cesaro bounded and P(d) = 0 for every d ∈ D, D ⊆ X dense=⇒ T mean ergodic.

Proposition

Let T = D or T = J and assume T : H∞υ (C)→ H∞υ (C) is continuous.TFAE:

(i) T : H∞υ (C)→ H∞υ (C) is uniformly mean ergodic,

(ii) T : H0υ(C)→ H0

υ(C) is uniformly mean ergodic,

(iii) limN→∞||T+···+TN ||υ

N = 0.

Moreover, if 1 ∈ συ(T ), then T is not uniformly mean ergodic.

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Page 15: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

HypercyclicityMean ergodicity

Mean ergodicity

Theorem (Lin)

Let T ∈ L(X ) such that ‖T n/n‖ → 0. Then,

T uniformly mean ergodic ⇐⇒ (I − T )X is closed .

Theorem (Lotz)

Let T ∈ L(H∞α ) such that ‖T n/n‖ → 0. Then,

T mean ergodic ⇐⇒ T uniformly mean ergodic .

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Page 16: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

HypercyclicityMean ergodicity

Mean ergodicity

Theorem

Let υ(r) = e−αr , r ≥ 0. D is power bounded if and only if α < 1. Itis uniformly mean ergodic on H∞α (C) and H0

α(C) if α < 1, not meanergodic if α > 1, and it is not mean ergodic on H∞1 (C) and notuniformly mean ergodic on H0

1 (C).

Let υ(r) = e−αr , r ≥ 0. J is never hypercyclic and it is powerbounded if and only if α ≥ 1. If α > 1, J is uniformly mean ergodicon H∞α (C) and H0

α(C) and it is not mean ergodic on these spaces ifα < 1. If α = 1, then J is not mean ergodic on H∞1 (C), and meanergodic but not uniformly mean ergodic on H0

1 (C).

For every weight υ, H is power bounded, not hypercyclic anduniformly mean ergodic on H∞υ (C).

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Page 17: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

HypercyclicityMean ergodicity

Summary

J 0 < α < 1 α = 1 α > 1Power bounded no yes yes

Hypercyclic on H0α no no no

Mean ergodic on H0α no yes yes

Mean ergodic on H∞α no no yesUniformly mean ergodic no no yes

D 0 < α < 1 α = 1 α > 1Power bounded yes no no

Hypercyclic on H0α no yes yes

Top. mixing on H0α no yes yes

Chaotic on H0α no no yes

Mean ergodic on H0α yes ? no

Mean ergodic on H∞α yes no noUniformly mean ergodic yes no no

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Page 18: Classical operators on weighted Banach spaces of entire functions · Classical operators on weighted Banach spaces of entire functions 13/21. Introduction Classical results Weighted

IntroductionClassical results

Weighted Banach spaces of holomorphic functionsContinuity, norms and spectrum

Dynamics of D and J on H∞a,α and H0

a,α

HypercyclicityMean ergodicity

Dynamics on weighted Bergman spaces

Given a weight υ, 1 ≤ p ≤ ∞, and 1 ≤ q <∞,

Bp,q(υ) :=

{f ∈ H(C) : ‖f ‖p,q,υ :=

(2π

∫ ∞

0

rυ(r)qMp(f , r)qdr

)1/q

<∞

}

Bp,∞(υ) := {f ∈ H(C) : ||f ||p,υ := supr>0

υ(r)Mp(f , r) <∞}

Bp,0(υ) := {f ∈ H(C) : limr→∞

υ(r)Mp(f , r) = 0}.

Mp(f , r) :=

(1

∫ 2π

0

|f (re it)|pdt)1/p

for 1 ≤ p <∞,

M∞(f , r) := sup|z|=r

|f (z)|, r ≥ 0.

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Appendix

Bibliography I

K.G. Grosse-Erdmann, A. Peris,Linear Chaos.Springer , Berlin, 2011.

A. Atzmon, B. Brive,Surjectivity and invariant subspaces of differential operators onweighted Bergman spaces of entire functions.Bergman spaces and related topics in complex analysis, 27–39,

Contemp. Math., 404, Amer. Math. Soc., Providence, RI, 2006.

M.J. Beltran, J. Bonet, C. Fernandez,Classical operators on weighted Banach spaces of entire functions.Proc.Amer.Math.Soc. To appear.

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Appendix

Bibliography II

J. Bonet,Dynamics of the differentiation operator on weighted spaces of entirefunctions.Math. Z. 261 (2009), 649-657.

A. Harutyunyan, W. Lusky,On the boundedness of the differentiation operator betweenweighted spaces of holomorphic functions.Studia Math. 184 (2008), 233–247.

W. Lusky,On the isomorphism classes of weighted spaces of harmonic andholomophic functions,Studia Math. 175 (2006) 19–45.

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Appendix

Bibliography III

M. Lin,On the Uniform Ergodic Theorem,Proc. Amer. Math. Soc. 43 (1974), 337–340.

H.P. Lotz,Tauberian theorems for operators on L∞ and similar spaces,Functional Analysis: Surveys and Recent Results III, K.D. Bierstedtand B. Fuchssteiner (Eds.), North Holland, Amsterdam, 1984, pp.117–133.

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