Weighted Composition Operators and Supercyclicity Criterion
We consider an equivalent condition to the property of Supercyclicity Criterion, and then we investigate this property for the adjoint of weighted composition operators acting on Hilbert spaces of analytic functions.
Let be a bounded linear operator on . For , the orbit of under is the set of images of under the successive iterates of : The vector is called supercyclic for if is dense in . Also a supercyclic operator is one that has a supercyclic vector. For some sources on these topics, see [1–16].
Let be a separable Hilbert space of functions analytic on a plane domain such that, for each in , the linear functional of evaluation at given by is a bounded linear functional on . By the Riesz representation theorem, there is a vector in such that . We call the reproducing kernel at .
A complex-valued function on is called a multiplier of if . The operator of multiplication by is denoted by and is given by .
If is a multiplier of and is a mapping from into , then by for every and is called a weighted composition operators.
The holomorphic self-maps of the open unit disk are divided into classes of elliptic and nonelliptic. The elliptic type is an automorphism and has a fixed point in . It is well known that this map is conjugate to a rotation for some complex number with . 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.
2. 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 be a separable Hilbert space and is a continuous linear mapping on . Suppose that there exist two dense subsets and in , a sequence of positive integers, and also there exist mappings such that(1) for every ,(2) for every and every .Then, is supercyclic.
If an operator holds in the assumptions of Theorem 2.1, then one says that satisfies the Supercyclicity Criterion.
Definition 2.2. Let be a bounded linear operator on a Hilbert space . We refer to as the generalized kernel of .
Theorem 2.3. Let be a bounded linear operator on a separable Hilbert space with dense generalized kernel. Then, the following conditions are equivalent:(1) has a dense range,(2) is supercyclic,(3) satisfies the Supercyclicity Criterion.
Proof. See [2, Corollary 3.3].
Remark 2.4. In , for the proof of implication of Theorem 2.3, it has been shown that is supercyclic which implies (by using Lemma 3.1 in ) that satisfies the Supercyclicity Criterion. This implication can be proved directly without using Lemma 3.1 in , as follows: If is a bounded linear operator on a separable Hilbert space with dense range and dense generalized kernel, then it follows that is supercyclic [1, Exercise 1.3]. Now suppose that is a supercyclic vector of . Set and the generalized kernel of . Since is supercyclic, there exist sequences , and such that and . Define by
Then, clearly, pointwise on and
for every and every . Hence, satisfies the Supercyclicity Criterion.
From now on let be a Hilbert space of analytic functions on the open unit disc such that contains constants and the functional of evaluation at is bounded for all in . Also let be a nonconstant multiplier of and let be an analytic map from into such that the composition operator is bounded on . We define the iterates ( times). By or we mean the th iterate of , hence for .
Definition 2.5. We say that is a B-sequence for if for all .
Corollary 2.6. Suppose that is a B -sequence for and has limit point in . If , then satisfies the Supercyclicity Criterion.
Proof. Put . Since , we get for all . Hence has dense generalized kernel. Now let for all , thus for all . This implies that is the zero constant function, because is nonconstant and has limit point in . Thus, has dense range and, by Theorem 2.3, the proof is complete.
Example 2.7. Let , , and define for all . Now by Corollary 2.6, the operator satisfies the Supercyclicity Criterion.
Theorem 2.8. Let be an elliptic automorphism with interior fixed point and satisfies the inequality for all in a neighborhood of the unit circle. Then, the operator satisfies the Supercyclicity Criterion.
Proof. Put and where Since is an automorphism with , thus is a rotation for some and every . Set and . Then, clearly , thus is similar to which implies that satisfies the Supercyclicity Criterion if and only if satisfies the Supercyclicity Criterion. Since when , so for all in a neighborhood of the unit circle. So, without loss of generality, we suppose that is a rotation and for all in a neighborhood of the unit circle. Therefore, there exist a constant and a positive number such that when , and when . Set and . Also, consider the sets where span is the set of finite linear combinations of . By using the Hahn-Banach theorem, and are dense subsets of . Since is a rotation, the sequence is a subset of the compact set for each in and . Now by, using the Banach-Steinhaus theorem, the sequence is bounded for each in and . Note that, for each , . So, if , then and if , then for each positive integer . Also, note that for every positive integer and (see ). Now, if , then as . Therefore the sequence converges pointwise to zero on the dense subset . Define a sequence of linear maps by extending the definition () linearly to . Note that, for all , the sequence is bounded and on which implies that is identity on the dense subset . Hence, for every and every . Now, by Theorem 2.1, the proof is complete.
Corollary 2.9. Under the conditions of Theorem 2.8, is supercyclic.
Proof. It is clear since satisfies the Supercyclicity Criterion.
Example 2.10. Let and . Then, the operator satisfies the Supercyclicity Criterion, because 0 is an interior fixed point of , and for .
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