eu ON THE HYPERCYCLICITY AND WEIGHTED COMPOSITION OPERATORS ON BANACH FUNCTION SPACES

In this paper we investigate the hypercyclicity of adjoint of a special weighted composition operators on a Banach function space. AMS Subject Classification: 47B37, 47B38


Introduction
Let B(X) denotes the set of all bounded linear operators on a Banach space X.An operator T ∈ B(X) is said to be hypercyclic if there exists a vector x ∈ X for which the orbit Orb(T, x) = {T n x : n ∈ N is dense in X and in this case we refer to x as a hypercyclic vector for T .The holomorphic self maps of the open unit disk U are divided into classes of elliptic and non-elliptic.The elliptic type is an automorphism and has a fixed point in U .It is well known that this map is conjugate to a rotation z → λz for some complex number λ with |λ| = 1.The maps of that are not elliptic are called of non-elliptic type.The iterate of a non-elliptic map can be characterized by the Grand Iteration Theorem ([3, p. 78]).By ψ ′ (w) we denote the angular derivative of ψ at w ∈ ∂U .Note that if w ∈ U , then ψ ′ (w) has the natural meaning of derivative.Also, by ψ n we mean the nth iteration of the function ψ.The unique attracting point w in the above proposition is called the Denjoy-Wolff point of ψ.
Consider the weighted composition operator C ϕ,ψ on a Banach space X of analytic functions defined by We will investigate the hypercyclicity of the operator C * ϕ,ψ .For some sources we refer to [1][2][3][4].

Main Results
Let X be a Banach space of analytic functions on the open unit disk U .For each λ ∈ U , the evaluation function e λ : X → C is defined by e λ (f ) = f (λ), f ∈ X.A complex valued function ϕ on U for which ϕX ⊆ X is called a multiplier of X.The set of all multipliers of X is denoted by M (X) and we have M (X) ⊆ H ∞ (U ).
It is well known that if the adjoint of a continuous operator T on a Banach space has an eigenvector, then T * fails to be hypercyclic (see [2]).
Lemma 2.1.Let X be a separable Banach space of analytic functions on the open unit disk U such that 1 ∈ X, and for each λ ∈ U , e λ ∈ X * .Also, let z ∈ M (X) and the map ϕ . By the Farrell-Rubel-Shields Theorem [1, Theorem 5.1, p.151], there is a sequence {p n } n of polynomials converging to f such that for all n, p n U ≤ c 0 for some c 0 > 0. So we obtain M pn = p n U ≤ c 0 for all n.But ball B(X) is compact in the weak operator topology and so by passing to a subsequence if necessary, we may assume that for some A ∈ B(X), M pn −→ A in the weak operator topology.Using the fact that M * pn −→ A * in the weak operator topology and acting these operators on e λ we get p n (λ)e λ = M * pn e λ −→ A * e λ weakly.Since p n (λ) −→ f (λ) we see that A * e λ = f (λ)e λ .Because the closed linear span of {e λ : λ ∈ U } is dense in X * , we conclude that A = M f and this implies that f ∈ M (X).Indeed f ∈ X, since X contains the constant functions.
Theorem 2.2.Let X be a separable Banach space of analytic functions on the open unit disk U such that 1 ∈ X, and for each λ ∈ U , e λ ∈ X * .Also, let z ∈ M (X) and the map ϕ → M ϕ from M (X) into B(X) is an isometry.Let 0 < λ < 1 and ϕ ∈ M (X) be such that ϕ(0) = 0. Then (C ϕ,λz ) * is not hypercyclic.
Proof.Since ϕ is bounded, an application of Schwartz's Lemma shows that By substituting λ i z instead of z in the above inequality, we get This implies that and consequently converges uniformly on compact subsets of U .Set then g is a nonzero holomorphic function on U .Also, note that Therefore, we get for every z ∈ U .Hence, indeed g belongs to H ∞ (U ) and so by Lemma 2.1, g ∈ X.But ϕ • g • λz = ϕ(0)g and so ϕ(0) is an eigenvalue for the operator M ϕ C ψ .Now since the operator C ϕ,λz has a non-zero eigenvalue, then (C ϕ,λz ) * fails to be hypercyclic.This completes the proof.
Received: December 24, 2014 c 2015 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author Proposition 1.1.(see [6]) Suppose ψ is a holomorphic self-map of U that is not an elliptic automorphism.If ψ has a fixed point w ∈ U , then ψ n →w uniformly on compact subsets of U and |ψ ′ (w)| < 1.