Q-Frequent hypercyclicity in spaces of operators

Abstract

We provide conditions for a linear map of the form $$C_{R,T}(S)=RST$$ 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 $$C_{R}(S)=RSR^*$$ is shown to be q-frequently hypercyclic on the space $$\mathcal {K}(H)$$ of all compact operators and the real topological vector space $$\mathcal {S}(H)$$ of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for $$C_{R,T}$$ to be q-frequently hypercyclic on the Schatten von Neumann classes $$S_p(H)$$ . We also characterize frequent hypercyclicity of $$C_{M^*_\varphi ,M_\psi }$$ on the trace-class of the Hardy space, where the symbol $$M_\varphi $$ denotes the multiplication operator associated to $$\varphi $$ .