## Journal of Generalized Lie Theory and Applications

### Existence Theorems in Linear Chaos

Stanislav Shkarin

#### Abstract

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.

#### Article information

Source
J. Gen. Lie Theory Appl., Volume 9, Number S1 (2015), 34 pages.

Dates
First available in Project Euclid: 11 November 2016

https://projecteuclid.org/euclid.jglta/1478833228

Digital Object Identifier
doi:10.4172/1736-4337.S1-009

Mathematical Reviews number (MathSciNet)
MR3637853

Zentralblatt MATH identifier
1371.37132

#### Citation

Shkarin, Stanislav. Existence Theorems in Linear Chaos. J. Gen. Lie Theory Appl. 9 (2015), no. S1, 34 pages. doi:10.4172/1736-4337.S1-009. https://projecteuclid.org/euclid.jglta/1478833228