Transitive and Hypercyclic Operators on Locally Convex Spaces

Abstract

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space ϕ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on ϕ). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

Existence and nonexistence of hypercyclic semigroups

“…In contrast, by Conejero [16] there do exist strongly continuous semigroups (T (t)) t≥0 on ϕ that are topologically transitive, that is, for any pair of nonempty open subsets U, V of ϕ there exists t ≥ 0 such that T (t)(U ) ∩ V = ∅. The analogous result for operators is due to Bonet, Frerick, Peris and Wengenroth [11].…”

Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

“…In [32] it was shown that = ⊕ ℓ , 1 ≤ < ∞, and = ⊕ = D(Ω), where is the space of rapidly decreasing functions and D(Ω) is the space of test functions on an open set Ω ⊂ R , support a noninjective hypercyclic operator. Shkarin [33] generalized this result to the present context by showing the existence of hypercyclic operators of the form = + for noninjective.…”

Dynamics on Binary Relations over Topological Spaces

“…It is well known that these notions are equivalent for continuous linear operators on Fréchet spaces by Baire's category theorem (see, for instance, [21,22]). However, there exist examples of continuous linear operators on non-metrizable locally convex spaces that are topologically transitive and not hypercyclic [23]. Moreover, for any non-trivial Banach space X there exists a multivalued linear operator A = {0} × X that is hypercyclic and not topologically transitive (cf.…”