Weighted Composition Operators and Supercyclicity Criterion

The vector x is called supercyclic for T ifC orb T, x is dense inH. Also a supercyclic operator is one that has a supercyclic vector. For some sources on these topics, see 1–16 . Let H be a separable Hilbert space of functions analytic on a plane domain G such that, for each λ in G, the linear functional of evaluation at λ given by f → f λ is a bounded linear functional on H. By the Riesz representation theorem, there is a vector Kλ in H such that f λ 〈f,Kλ〉. We call Kλ the reproducing kernel at λ. A complex-valued function φ on G is called a multiplier ofH if φH ⊂ H. The operator of multiplication by φ is denoted byMφ and is given by f → φf . If φ is a multiplier ofH and ψ is a mapping from G into G, then Cφ,ψ : H → H by


Introduction
Let T be a bounded linear operator on H.For x ∈ H, the orbit of x under T is the set of images of x under the successive iterates of T : orb T, x x, T x, T 2 x, . . . .

1.1
The vector x is called supercyclic for T if C orb T, x is dense in H. Also a supercyclic operator is one that has a supercyclic vector.For some sources on these topics, see 1-16 .
Let H be a separable Hilbert space of functions analytic on a plane domain G such that, for each λ in G, the linear functional of evaluation at λ given by f → f λ is a bounded linear functional on H.By the Riesz representation theorem, there is a vector K λ in H such that f λ f, K λ .We call K λ the reproducing kernel at λ.A complex-valued function ϕ on G is called a multiplier of H if ϕH ⊂ H.The operator of multiplication by ϕ is denoted by M ϕ and is given by f → ϕf.
If ϕ is a multiplier of H and ψ is a mapping from G into G, then C ϕ,ψ : H → H by for every f ∈ H and z ∈ G is called a weighted composition operators.

International Journal of Mathematics and Mathematical Sciences
The holomorphic self-maps of the open unit disk D are divided into classes of elliptic and nonelliptic.The elliptic type is an automorphism and has a fixed point in D. 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 those which are not elliptic are called of non-elliptic type.The iterate of a non-elliptic map can be characterized by the Denjoy-Wolff Iteration theorem.

Main Results
We will investigate the property of Hypercyclicity Criterion for a linear operator and in the special case, we will give sufficient conditions for the adjoint of a weighted composition operator associated with elliptic composition function which satisfies the Supercyclicity Criterion.
Theorem 2.1 Supercyclicity Criterion .Let H be a separable Hilbert space and T is a continuous linear mapping on H. Suppose that there exist two dense subsets Y and Z in H, a sequence {n k } of positive integers, and also there exist mappings Then, T is supercyclic.
If an operator T holds in the assumptions of Theorem 2.1, then one says that T satisfies the Supercyclicity Criterion.Definition 2.2.Let T be a bounded linear operator on a Hilbert space H.We refer to n≥1 Ker T n as the generalized kernel of T .Theorem 2.3.Let T be a bounded linear operator on a separable Hilbert space H with dense generalized kernel.Then, the following conditions are equivalent: Remark 2.4.In 2 , for the proof of implication 1 → 3 of Theorem 2.3, it has been shown that T T is supercyclic which implies by using Lemma 3.1 in 2 that T satisfies the Supercyclicity Criterion.This implication can be proved directly without using Lemma 3.1 in 2 , as follows: If T is a bounded linear operator on a separable Hilbert space H with dense range and dense generalized kernel, then it follows that T is supercyclic 1, Exercise 1.3 .Now suppose that h 0 is a supercyclic vector of T .Set X 0 C orb T, h 0 and Y 0 the generalized kernel of T .Since T is supercyclic, there exist sequences {n j } j ⊂ N, {α j } j ⊂ C and {f j } j ⊂ H such that f j → 0 and α j T n j f j → h 0 .Define S n k : X 0 → H by International Journal of Mathematics and Mathematical Sciences 3 Then, clearly, T n k S n k → I X 0 pointwise on X 0 and for every y ∈ Y 0 and every x ∈ X 0 .Hence, T satisfies the Supercyclicity Criterion.
From now on let H be a Hilbert space of analytic functions on the open unit disc D such that H contains constants and the functional of evaluation at λ is bounded for all λ in D. Also let ϕ : D → C be a nonconstant multiplier of H and let ψ be an analytic map from D into D such that the composition operator C ψ is bounded on H.We define the iterates Corollary 2.6.Suppose that {z n } n≥0 ⊂ D is a B -sequence for ψ and has limit point in D. If ϕ z 0 0, then C * ϕ,ψ satisfies the Supercyclicity Criterion.
Proof.Put A C ϕ,ψ .Since ϕ z 0 0, we get K z i ∈ Ker A * n for all i 0, ..., n − 1. Hence A * has dense generalized kernel.Now let f, A * K z n 0 for all n, thus ϕ z n • f • ψ z n 0 for all n.This implies that f is the zero constant function, because ϕ is nonconstant and {z n } n≥0 has limit point in D .Thus, A * has dense range and, by Theorem 2.3, the proof is complete.
Since Ψ is an automorphism with Ψ 0 0, thus Ψ is a rotation z → e iθ z for some θ ∈ 0, 2π and every z ∈ U. Set T C * Φ,Ψ and S C * ϕ,ψ .Then, clearly S * C α p T * C −1 α p , thus T is similar to S which implies that S satisfies the Supercyclicity Criterion if and only if T satisfies the Supercyclicity Criterion.Since all z in a neighborhood of the unit circle.So, without loss of generality, we suppose that ψ is a rotation z → e iθ z and |ϕ 0 | < 1 ≤ |ϕ z | for all z in a neighborhood of the unit circle.Therefore, there exist a constant λ and a positive number δ < 1 such that |ϕ z | < λ < 1 when |z| < δ, and |ϕ z | ≥ 1 when |z| > 1 − δ.Set U 1 {z : |z| < δ} and U 2 {z : |z| > 1 − δ}.Also, consider the sets where span {•} is the set of finite linear combinations of {•}.By using the Hahn-Banach theorem, H 1 and H 2 are dense subsets of H. Since ψ is a rotation, the sequence {ψ m n λ } n is a subset of the compact set {z : |z| λ} for each λ in D and m −1, 1.Now by, using the Banach-Steinhaus theorem, the sequence {K ψ m n λ } n is bounded for each λ in D and m −1, 1.Note that, for each Z, for every positive integer n and z ∈ D see 12 .Now, if z ∈ U 1 , then S n K z → 0 as n → ∞.
Therefore the sequence {S n } converges pointwise to zero on the dense subset H 1 .Define a sequence of linear maps W n : H 2 → H 2 by extending the definition

Example 2 . 7 .Theorem 2 . 8 .
Let ψ z e − √ 2πi z, ϕ z z − 1/2 , and define z n 1/2 e √ 2nπi for all n ≥ 0. Now by Corollary 2.6, the operator C * ϕ,ψ satisfies the Supercyclicity Criterion.Let ψ be an elliptic automorphism with interior fixed point p and ϕ : D → C satisfies the inequality |ϕ p | < 1 ≤ |ϕ z | for all z in a neighborhood of the unit circle.Then, the operator C * ϕ,ψ satisfies the Supercyclicity Criterion.Proof.Put Ψ α p • ψ • α p and Φ ϕ • α p where 6z ∈ U 2 linearly to H 2 .Note that, for all z ∈ U 2 , the sequence {W n K z } n is bounded and S n W n K z K z on H 2 which implies that S n W n is identity on the dense subset H 2 .Hence,S n f W n g −→ 0 2.7for every f ∈ H 1 and every g ∈ H 2 .Now, by Theorem 2.1, the proof is complete.It is clear since C * ϕ,ψ satisfies the Supercyclicity Criterion.Example 2.10.Let ϕ z 3/2 z and ψ z e iθ z.Then, the operator C * ϕ,ψ satisfies the Supercyclicity Criterion, because 0 is an interior fixed point of ψ, and ϕ 0 < 1 ≤ |ϕ z | for |z| > 2/3.