$q$-Frequent hypercyclicity in spaces of operators

TL;DR

This paper establishes conditions under which certain linear operator maps are $q$-frequently hypercyclic on various spaces of operators, extending hypercyclicity concepts to operator algebras on Banach and Hilbert spaces.

Contribution

It introduces new criteria for $q$-frequent hypercyclicity of operator maps, including specific cases on compact, self-adjoint, and Schatten classes, and characterizes hypercyclicity on Hardy space trace-class.

Findings

Operators satisfying the $q$-Frequent Hypercyclicity Criterion are $q$-frequently hypercyclic.

The map $C_{R}(S)=RSR^*$ is $q$-frequently hypercyclic on compact and self-adjoint operators.

Conditions for $C_{R,T}$ to be $q$-frequently hypercyclic on Schatten classes.

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 $\varphi$.