Hypercyclicity of the adjoint of weighted composition operators
type: Item
instance of: scholarly article
title: Hypercyclicity of the adjoint of weighted composition operators
zbMATH Open document ID: 1191.47008
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2009-12-09T00:00:00Z
review text: 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.
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 5644691
zbMATH Keywords: weighted composition operator
zbMATH Keywords: hypercyclic operator
zbMATH Keywords: Denjoy-Wolff point
zbMATH Keywords: loxodromic map
zbMATH Keywords: Farrell-Rubel-Shields theorem
author: Bahmann Yousefi
author: Rahmat Soltani
MaRDI profile type: Publication
full work available at URL: s12044-009-0048-6
OpenAlex ID: W2069729755
cites work: Cyclic phenomena for composition operators
cites work: Operators with dense, invariant, cyclic vector manifolds
cites work: Q3786684
cites work: Composition operators and classical function theory
cites work: Q5607489
cites work: Hypercyclic property of weighted composition operators
cites work: Hypercyclicity of special operators on Hilbert function spaces
DOI: 10.1007/S12044-009-0048-6
description: scientific article; zbMATH DE number 5644691