Spectral characterization of weak topological transitivity

Abstract

Given a bounded linear operator S on a real Banach space X , we characterize weak topological transitivity of the operator families {S t | t ∈ N}, {κ S t | t ∈ N, κ > 0}, and {κ S t | t ∈ N, κ ∈ R} in terms of the point spectrum of the dual operator S * .

A review of some recent work on hypercyclicity

“…for all disjoint nonempty open subsets U, V of M there is a number n such that U T −n V contains a nonempty open set of M. Le [77] gave a sufficient condition for M -hypercyclicity and used it to show that it need not imply M -transitivity. Desch and Schappacher [45] defined the (weakly) topological transitivity of a semigroup S of bounded linear operators on a real Banach space as the property for all nonempty (weakly) open sets U, V that for some T ∈ S we have T U V = ∅. They characterized weak topological transitivity of the families of operators {S t |t ∈ N}, {kS t |t ∈ N, k > 0}, and {kS t |t ∈ N, k ∈ R}, in terms of the point spectrum of the dual operator S * cf.…”

“…also [9]. Unlike topological transitivity in the norm topology, which is equivalent to hypercyclicity with concomitant highly irregular behaviour of the semigroup, Desch and Schappacher [45] illustrated quite good behaviour of weakly topologically transitive semigroups. They gave an example using the positive-definite bounded self-adjoint…”