Existence theorems in linear chaos

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dc.creatorShkarin, S.
dc.date2008-10-07
dc.date2008-10-22
dc.date.accessioned2026-07-07T10:11:59Z
dc.date.available2026-07-07T10:11:59Z
dc.descriptionChaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. 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 $k\in\N$, any separable infinite dimensional Fréchet space $X$ non-isomorphic to $ω$ admits a mixing uniformly continuous group $\{T_t\}_{t\in C^n}$ of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup $\{T_t\}_{t\geq 0}$ 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.
dc.identifierhttps://arxiv.org/abs/0810.1192
dc.identifierhttp://arxiv.org/abs/0810.1192
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/172036
dc.subjectFunctional Analysis
dc.subjectDynamical Systems
dc.subject47A16; 37A25
dc.titleExistence theorems in linear chaos
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