Abstract
We give sufficient conditions under which a weighted composition operator on a Hilbert space of analytic functions is not weakly supercyclic. Also, we give some necessary and sufficient conditions for hypercyclicity and supercyclicity of weighted composition operators on the space of analytic functions on the open unit disc.
Highlights
Let D denote the open unit disc in the complex plane
An application of the closed graph theorem shows that the weighted composition operator Cw,φ defined by Cw,φ f z MwCφ fzwzfφz is bounded
The following proposition limits the kinds of maps that can produce weakly supercyclic weighted composition operators on H
Summary
Let H be a Hilbert space of analytic functions defined on D such that 1, z ∈ H, and for each λ in D, the linear functional of point evaluation at λ given by f → f λ is bounded. A well-known example of a Hilbert space of analytic functions is the weighted Hardy space. The weighted Hardy space H2 β is defined as the space of functions f. An application of the closed graph theorem shows that the weighted composition operator Cw,φ defined by Cw,φ f z MwCφ fzwzfφz is bounded. The hypercyclicity of the adjoint of a weighted composition operator on a Hilbert space of analytic functions is discussed in. We discuss weak supercyclicity and weak hypercyclicity of a weighted composition operator
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