Existence theorems in linear chaos

Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat the questions of the following type. Characterize which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize LBS-spaces (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 Fréchet space X if and only if X is non-isomorphic to the space ω of all sequences with pointwise convergence topology. It is also shown that a separable infinite dimensional Fréchet space X admits a mixing strongly continuous semigroup {Tt}t>0 of continuous linear operators if and only if X is non-isomorphic to 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. MSC: 47A16, 37A25