The dynamic behavior of conjugate multipliers on some reflexive Banach spaces of analytic functions

Extending previous results of Godefroy and Shapiro we characterize the hypercyclic, mixing and chaotic conjugate multipliers on some reflexive spaces of analytic functions.

A continuous linear operator T on a Banach space X is called hypercyclic if there is an element x in X whose orbit {T n x : n ∈ N} under T is dense in X; topologically transitive if for any pair U, V of nonempty open subsets of X, there exists some nonnegative integer n such that T n (U) ∩ V = ∅; mixing if for any pair U, V of nonempty open subsets of X, there exists some nonnegative integer N such that T n (U) ∩ V = ∅ for all n N; and chaotic if T is topologically transitive and T has a dense set of periodic points.
The historical interest in hypercyclicity is related to the invariant subset problem.The invariant subset problem, which is open to this day, asks whether every continuous linear operator on any infinite dimensional separable Hilbert space possesses an invariant closed subset other than the trivial ones given by {0} and the whole space.Counterexamples do exist for continuous linear operators on non-reflexive spaces like l 1 .After a simple observation, a continuous linear operator T on a Banach space X has no nontrival invariant closed subsets if and only if every nonzero vector x is hypercyclic (i.e. the orbit {T n x : n ∈ N} under T is dense in X).
The best known examples of hypercyclic operators are due to Birkhoff [4], MacLane [15] and Rolewicz [18].Each of these papers had a profound influence on the literature on hypercyclicity.Birkhoff's result on the hypercyclicity of the translation operator T a (f )(z) = f (z + a), a = 0, on the space H(C) of entire functions has led to an extensive study of hypercyclic composition operators (see [12, pages 110-118]), while MacLane's result on the hypercyclicity of the differentiation operator Df = f ′ on H(C) initiated the study of hypercyclic differential operators (see [12, pages 104-110]).
Recently Godefroy and Shapiro [10] have studied the dynamic properties of conjugate multipliers on some Hilbert spaces of analytic functions.Godefroy and Shapiro [10] characterized hypercyclic, mixing and chaotic conjugate multipliers on some Hilbert spaces of analytic functions.It is therefore very natural to try to characterize hypercyclic, mixing and chaotic conjugate multipliers on arbitrary reflexive Banach spaces of analytic functions.In this paper we will characterize the hypercyclic, mixing and chaotic conjugate multipliers on some reflexive spaces of analytic functions, generalizing [10, Theorem 4.5, Theorem 4.9, Theorem 6.2].
Theorem 1.1.Let Ω ⊆ C be a nonempty open connected subset.Let X = {0} be a reflexive Banach space of analytic functions on Ω such that each point evaluation Let ϕ be a nonconstant bounded analytic function on Ω and M * ϕ the conjugate of M ϕ .Then the following assertions are equivalent: (1) Godefroy and Shapiro [10, Theorem 4.5, Theorem 4.9, Theorem 6.2] proved Theorem 1.1 in the case of Hilbert spaces of analytic functions, thus Theorem 1.1 generalizes [10, Theorem 4.5, Theorem 4.9, Theorem 6.2].
This paper is organized as follows.In Section 2 we characterize the hypercyclic, mixing and chaotic conjugate multipliers on some reflexive spaces of analytic functions, generalizing [10, Theorem 4.5, Theorem 4.9, Theorem 6.2].Furthermore, we exhibit several hypercyclic, mixing and chaotic conjugate multipliers on H p spaces for p > 1.These examples show that our generalizations are more effective.Acknowledgments.Z. R. was supported by National Natural Science Foundation of China (Grant No.12261063).

The dynamic behavior of conjugate multipliers on some reflexive Banach spaces of analytic functions
In this section we characterize hypercyclic, mixing and chaotic conjugate multipliers on some reflexive Banach spaces of analytic functions, generalizing [10, Theorem 4.5, Theorem 4.9, Theorem 6.2].

Recall the notion of annihilator introduced in [21, page 163].
Definition 2.1.Let X be a normed linear space.If A ⊆ X, the annihilator A ⊥ of A is the set where X * is the set of continuous linear functionals on X.
If F ⊆ X * , the annihilator F ⊥ of F is the set The following technical results will help us characterize hypercyclic, mixing and chaotic conjugate multipliers on some reflexive Banach spaces of analytic functions.
Proposition 2.2.A normed linear space X is a reflexive Banach space if and only if every norm-closed linear subspace in X * is σ(X * , X)-closed, where σ(X * , X) is the weak * topology on X * .
Proposition 2.3.Let X be a normed linear space.If F is a nonempty subset of X * , then F ⊥⊥ is the σ(X * , X)-closed linear subspace generated by F , where The following proposition is well known in complex analysis (see [6, page 78]).(1) f ≡ 0; (2) {z ∈ Ω : f (z) = 0} has a limit point in Ω.
Proposition 2.5.Let T be a continuous linear operator on a separable Banach space X.Suppose that the subspaces are dense in X.Then T is mixing, and in particular hypercyclic.If, moreover, X is a complex space and also the subspace The following is the major technique we need.
Lemma 2.6.Let Ω ⊆ C be a nonempty open connected subset.Let X = {0} be a reflexive Banach space of analytic functions on Ω such that each point evaluation Let Λ ⊆ Ω be a set with a limit point in Ω.Then the set span{k λ : λ ∈ Λ} is dense in X * .
Next we will show that is a root of unity} has a limit point in Ω, by Lemma 2.6 we have span{k λ : λ ∈ Ω and ϕ(λ) is a root of unity} is dense in X * .Hence C is dense in X * .By (4)⇒(2), we have M * ϕ is mixing.By Proposition 2.5, we have M * ϕ is chaotic.
(3)⇒(1) Assume that M * ϕ is chaotic.Then M * ϕ is topologically transitive.Since a continuous linear operator on a separable Banach space is topologically transitive if and only if it is hypercyclic (see [12,  Then (H p , • p ) is a Banach space.
In this example we will characterize hypercyclic, mixing and chaotic conjugate multipliers on H p for 1 < p < +∞, generalizing [10, page 253].

1 .
Introduction Throughout this article, let N denote the set of nonnegative integers.Let K denote the real number field R or the complex number field C. Let Q denote the rational number field.If z ∈ C and r > 0 are fixed then define B(z, r) = {λ ∈ C : |λ−z| < r}.Let T = {z ∈ C : |z| = 1} and D = {z ∈ C : |z| < 1}.

1 p
page 10]), by Remark 2.7 we have M * ϕ is hypercyclic.✷ Godefroy and Shapiro [10, Theorem 4.5, Theorem 4.9, Theorem 6.2] proved Theorem 1.1 in the case of Hilbert spaces of analytic functions, thus Theorem 1.1 generalizes [10, Theorem 4.5, Theorem 4.9, Theorem 6.2].Example 2.8.For 1 p < +∞, let H p denote the space of all analytic functions on D for which sup (re iθ )| p dθ) < +∞.For any f ∈ H p , letf p = sup λ). Next we will show that every bounded analytic function ψ on D defines a multiplication operator M ψ : H p → H p with M ψ sup z∈D |ψ(z)| for 1 p < +∞.Let ψ be a bounded analytic function on D. Then ψ defines a multiplication operator M ψ on H p .Furthermore, for any f ∈ H p we have M ψ (f ) p = sup iθ )f (re iθ )| p dθ)
p ) * and (H p , • p ) is reflexive.By Theorem 1.1, for any nonconstant bounded analytic function ϕ on D and 1 < p < +∞, M * Godefroy and Shapiro [10, page 253] proved the above result in the case of p = 2, thus the above example generalizes [10, page 253].