q-Frequent hypercyclicity in spaces of operators

Abstract

Abstract. We provide conditions for a linear map of the form CR,T (S) = RST to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-Frequent Hypercyclicity Criterion, then the map CR(S)=RSR * is shown to be q-frequently hypercyclic on the space K(H) of all compact operators and the real topological vector space S(H) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for CR,T to be q-frequently hypercyclic on the Schatten von Neumann classes Sp(H). We also characterize frequent hypercyclicity of CM * ϕ ,M ψ on the trace-class of the Hardy space, where the symbol Mϕ denotes the multiplication operator associated to ϕ.

F-hypercyclic operators on Fréchet spaces

“…The notion of an (m n )-hypercyclic operator on a separable Fréchet space was introduced by Bayart and Matheron [4] in 2009 with a view to control the frequency of the individual orbits of a hypercyclic, non-weakly mixing operator. Gupta, Mundayadan [20]- [21] and Heo, Kim-Kim [22] have been recently considered the special case of (m n )-hypercyclicity, the so-called q-frequent hypercyclicity, where the sequence (m n ) is given by m n := n q (q ∈ N). In the case that q = 1, the q-frequent hypercyclicity is also known as frequent hypercyclicity and, without any doubt, that is the best explored concept of above-mentioned.…”