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2015 Existence Theorems in Linear Chaos
Stanislav Shkarin
J. Gen. Lie Theory Appl. 9(S1): 1-34 (2015). DOI: 10.4172/1736-4337.S1-009


Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. In this survey paper, we treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property.

In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fréchet space $X$ if and only if $X$ is non-isomorphic to the space $ω$ of all sequences with coordinatewise convergence topology. It is also shown for any $k ∈ \mathbb{N}$, any separable infinite dimensional Fréchet space $X$ non-isomorphic to $ω$ admits a mixing uniformly continuous group $\{T_t\}_{t∈C^n}$ of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup $\{T_t\}_{t≥0}$ on $ω$. We specify a wide class of Fréchet spaces $X$, including all infinite dimensional Banach spaces with separable dual, such that there is a hypercyclic operator $T$ on $X$ for which the dual operator $T′$ is also hypercyclic. An extension of the Salas theorem on hypercyclicity of a perturbation of the identity by adding a backward weighted shift is presented and its various applications are outlined.


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Stanislav Shkarin. "Existence Theorems in Linear Chaos." J. Gen. Lie Theory Appl. 9 (S1) 1 - 34, 2015.


Published: 2015
First available in Project Euclid: 11 November 2016

zbMATH: 1371.37132
MathSciNet: MR3637853
Digital Object Identifier: 10.4172/1736-4337.S1-009

Keywords: Backward weighted shifts , Bilateral weighted shifts , Hypercyclic operators , Mixing semigroups

Rights: Copyright © 2015 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

Vol.9 • No. S1 • 2015
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