$q$-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 $C_R(S)$=$RS{R}^{\ast }$ 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 $C_{M_{\phi }^{\ast },M_{\psi }}$ on the trace-class of the Hardy space, where the symbol $M_{\phi }$ denotes the multiplication operator associated to $\phi$.

The previous version has been changed considerably with many corrections rectified