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Hypercyclicity of the adjoint of weighted composition operators. (English) Zbl 1191.47008
A bounded linear operator $$T$$ on a Banach space $$X$$ is said to be hypercyclic if there exists a vector $$x\in X$$ such that the orbit $$\{ T^n x: n=0,1,2,\dots\}$$ is dense in $$T$$. Let $$X$$ be a Banach space of analytic functions on the open unit disk $$U$$. Assume that an analytic function $$\psi:U \to U$$ induces a bounded composition operator $$C_{\psi}$$ on $$X$$ and that an analytic function $$\varphi$$ on $$U$$ induces a bounded multiplication operator $$M_{\varphi}$$ on $$X$$. Let $$A=M_{\varphi}C_{\psi}$$. It is shown that, under some conditions on $$X$$, $$\psi$$, and $$\varphi$$, the adjoint operator $$A^{*}$$ cannot be hypercyclic.
The main idea in the proofs is showing that $$A$$ has an eigenvalue. Namely, it is well-known that the adjoint of an operator with an eigenvalue cannot be hypercyclic.

##### MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators 47B33 Linear composition operators
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##### References:
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