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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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