$q$-Frequent hypercyclicity in spaces of operators

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 $\varphi$.