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 Frechet 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 ∈ N, any separable infinite dimensional Frechet space X non-isomorphic to ω admits a mixing uniformly continuous group {Tt}t∈Cn T of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup {Tt}t≥0 on ω. We specify a wide class of Frechet 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.
Highlights
Notation and definitionsA topological vector space is called locally convex if it has a base of neighborhoods of zero consisting of convex sets
There are no hypercyclic operators on any finite dimensional topological vector space and there are no supercyclic operators on a finite dimensional topological vector space of real dimension > 2
Herzog [8] demonstrated that there is a supercyclic operator on any separable infinite dimensional Banach space
Summary
A topological vector space is called locally convex if it has a base of neighborhoods of zero consisting of convex sets. Let T be a continuous linear operator on a topological vector space X. If A is a normed semigroup and {Tt}t∈A is an operator semigroup on a topological vector space X, we say that {Tt}t∈A is (topologically) transitive if for any non-empty open subsets U, V of X, the set {|t| : t ∈ A, Tt(U)∩V ≠∅} is unbounded. A continuous linear operator T acting on a topological vector space X is called hypercyclic, supercyclic, hereditarily hypercyclic, hereditarily supercyclic, mixing or transitive if the semigroup {Tn} n∈ + has the same property. If {Tt}t∈A is mixing, Tt is mixing whenever |t| > 0
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