eu DYNAMICS OF A SPECIAL COMBINATION OF WEIGHTED COMPOSITION OPERATORS ON HILBERT FUNCTION SPACES

In this paper we give some sufficient conditions for the adjoint of a combination of weighted composition operators, acting on some function spaces, satisfying the hypercyclicity criterion. AMS Subject Classification: 47B37, 47B33


Introduction
The holomorphic self maps of the open unit disk D are divided into classes of elliptic and non-elliptic.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.A non-elliptic holomorphic self-map of D with Denjoy-Wolff point in D is called a dilation type.
Let H be a Hilbert space of functions analytic on the open unit disc D such that for each λ in D the linear functional of evaluation at λ given by f −→ f (λ) is a bounded linear functional on H.
A complex-valued function ψ on D is called a multiplier of a Hilbert space H if ψH ⊂ H.The operator of multiplication by ψ is denoted by M ψ and is given by f −→ ψf .
If w is a multiplier of H and ϕ is a mapping from D into D such that f • ϕ ∈ H for all f ∈ H, then C ϕ (defined on H by C ϕ f = f • ϕ) and M w C ϕ are called composition and weighted composition operator, respectively.
By an n-tuple of operators we mean a finite sequence of length n of commuting continuous linear operators on a locally convex space X.If T = (T 1 , T 2 , ..., T n ) is an n-tuple of operators, then we will let be the semigroup generated by T .For x ∈ X, the orbit of x under the tuple T is the set orb(T, x) = {Sx : S ∈ F}.A vector x is called a hypercyclic vector for T if orb(T, x) is dense in X and in this case the tuple T is called hypercyclic.The vector x is called supercyclic for T if Corb(T, x) is dense in X.Also a supercyclic tuple is one that has a supercyclic vector.When n = 1, then orbits of single operators have been studied widely.For some works on the topics of hypercyclicity and supercyclicity we refer to [1 -4].

Main Results
In this section we give some sufficient conditions for the adjoint of a combination of weighted composition operators, acting on a Hilbert space of analytic functions, satisfying the hypercyclicity criterion.First, we state the hypercyclicity criterion in the weak form due to Gethner and Shapiro ( [2]): Theorem 2.1.(Hypercyclicity Criterion) Let X be a separable Frechet space and T a continuous linear operator on X.If there exist dense subsets X 0 , Y 0 of X, an increasing sequence {n k } k of positive integers and mapping S : Y 0 −→ X such that T n k x −→ 0 and S n k y −→ 0 for every x ∈ X 0 , y ∈ Y 0 , and T • S = I Y 0 , the identity on Y 0 , then the operator T is hypercyclic.
In the following, we give the hypercyclicity criterion for a 2-tuples that can be extended similarly to any k-tuple.Theorem 2.2.(The Hypercyclicity Criterion for Tuples) Suppose X is a separable Banach space and T = (T 1 , T 2 ) is a pair of continuous linear mappings on X.If there exist two dense subsets Y and Z in X and two strictly increasing sequences {n j } and {k j } such that: 2 y → 0 for every y ∈ Y , and 2. There exist functions S j : Z → X such that for every z ∈ Z, S j z → 0, and T As earlier, we suppose that H is a separable 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, and H is automorphism invariant.The proof of the following theorem follows by the same method used in the proof of Theorem 3.3 in [4].

Then the adjoint of the operator
Proof.Without loss of generality suppose that p = 0. Then Therefore, there exist a constant λ and a positive number δ < 1 such that and H 2 = span{k z : |z| > 1 − δ}, where span {.} is the set of finite linear combinations of {.}.So H 1 and H 2 are dense subsets of H.
Define the linear map : for every z ∈ D. Now by the same method used in [4] we can see that: L n k z → 0, S n k y → 0, LS(k y ) = k y for all k z ∈ H 1 and k y ∈ H 2 .Therefore L satisfies the hypothesis of hypercyclicity criterion and so the proof is complete.
Corollary 2.4.Under the conditions of Theorem 2.3, if Proof.First note that by the assumption ϕ i ,ψ i commutes with C * ϕ j ,ψ j for all i, j ∈ {1, ..., k}, and indeed T is a ktuple.Now clearly the proof is complete.
Proof.For all n i ∈ N (i = 1, .., k), set if there exists j ∈ {1, 2, ..., k} such that ϕ j (z) = 0 for some z ∈ D, then C * ϕ j ,ψ j k z = ϕ j (z)k ψ j (z) = 0.This implies that L n (k z ) = 0 that is a contradiction since it is well-known that every nonzero orbit of the adjoint of a hypercyclic tuple should be unbounded.Also, note that ϕ i (p) k p , and so k i=1 ϕ i (p) is an eigenvalue of M * , as desired.But it is well known that the adjoint of a hypercyclic operatorcan not have an eigenvector, hence the operator M fails to be hypercyclic and so the proof is complete.