A hypercyclicity criterion with applications
type: Item
instance of: scholarly article
title: A hypercyclicity criterion with applications
zbMATH Open document ID: 1108.47009
author: Henrik Petersson
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: The following hypercyclicity criterion for continuous linear operators \(T: X \rightarrow X\) defined on a Hausdorff locally convex space \((X,\tau)\) is proved: Assume that \(X\) admits a finer topology \(\mu\) such that \((X,\mu)\) is a Fréchet space and \(T\) is \(\mu\)-continuous. Suppose that there is a countable \(\tau\)-dense subset \(Y\) of \(X\) and there is a sequence \((S_n)_n\) of maps \(S_n:Y \rightarrow X\) such that, for all \(y \in Y\), we have \(T_n(y) \rightarrow 0\), \(S_n(y) \rightarrow 0\) in \((X,\mu)\) and \(T^n S_n(y)=y\). Then the operator \(T\) is hypercyclic for \((X,\tau)\), i.e., there is \(x \in X\) such that its orbit \(\{x, T(x), T^2(x),\dots\}\) is dense in \((X,\tau)\). The proof is constructive and does not require that \((X,\mu)\) is separable. The criterion is used to show weak-* hypercyclicity of weighted backward shift operators on the dual of Banach sequence spaces. As the main application, the author exhibits hypercyclic left multipliers \(L_T: S \rightarrow TS\) on spaces of operators \(L(X,Y)\), endowed with the topology of uniform convergence on bounded sets, for several pairs \((X,Y)\) of spaces of holomorphic functions of several variables. It is also investigated when \(L_T\) admits a complemented infinite-dimensional subspace of hypercyclic vectors.
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 5083361
zbMATH Keywords: hypercyclic operators
zbMATH Keywords: hypercyclicity criterion
zbMATH Keywords: backward shift
zbMATH Keywords: left multiplier
reviewed by: José Bonet
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.05.019
OpenAlex ID: W2028041765
cites work: Hereditarily hypercylic operators
cites work: Invariant manifolds of hypercyclic vectors for the real scalar case
cites work: Universal and chaotic multipliers on spaces of operators
cites work: Hypercyclic operators on non-normable Fréchet spaces
cites work: Q4529373
cites work: Q5705641
cites work: Hypercyclic subspaces of a Banach space
cites work: Operators with dense, invariant, cyclic vector manifolds
cites work: Q4136031
cites work: CHAOS FOR BACKWARD SHIFT OPERATORS
cites work: Banach spaces of hypercyclic vectors
cites work: PDE-PRESERVING PROPERTIES
cites work: Q5490312
cites work: Hypercyclic conjugate operators
cites work: Q5294005
cites work: Hypercyclic sequences of PDE-preserving operators
cites work: Supercyclic and Hypercyclic Non-Convolution Operators on Hilbert–Schmidt Enire Functions
cites work: Hypercyclic operators on non-locally convex spaces
DOI: 10.1016/J.JMAA.2006.05.019
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description: scientific article; zbMATH DE number 5083361