Post on 22-Jul-2020
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
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.
Classical operators on weighted Banach spaces of entire functions 2/21
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 .
Classical operators on weighted Banach spaces of entire functions 3/21
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.
Classical operators on weighted Banach spaces of entire functions 4/21
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.
Classical operators on weighted Banach spaces of entire functions 5/21
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.
Classical operators on weighted Banach spaces of entire functions 6/21
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.
Classical operators on weighted Banach spaces of entire functions 7/21
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,∞[.
Classical operators on weighted Banach spaces of entire functions 8/21
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.
Classical operators on weighted Banach spaces of entire functions 9/21
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).
Classical operators on weighted Banach spaces of entire functions 10/21
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
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).
Classical operators on weighted Banach spaces of entire functions 12/21
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.
Classical operators on weighted Banach spaces of entire functions 13/21
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.
Classical operators on weighted Banach spaces of entire functions 14/21
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 .
Classical operators on weighted Banach spaces of entire functions 15/21
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).
Classical operators on weighted Banach spaces of entire functions 16/21
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
Classical operators on weighted Banach spaces of entire functions 17/21
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
∫ 2π
0
|f (re it)|pdt)1/p
for 1 ≤ p <∞,
M∞(f , r) := sup|z|=r
|f (z)|, r ≥ 0.
Classical operators on weighted Banach spaces of entire functions 18/21
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.
Classical operators on weighted Banach spaces of entire functions 19/21
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.
Classical operators on weighted Banach spaces of entire functions 20/21
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.
Classical operators on weighted Banach spaces of entire functions 21/21