Existence Theorems in Linear Chaos

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.