Subspace-hypercyclicity of conditional weighted translations on locally compact groups

Abstract

Let G be a second countable locally compact group, $\mathcal {B}$ a Borel $\sigma$-algebra and let v be a Borel measurable weight function on G. In this paper, we study the subspace-hypercyclicity of the conditional weighted translation $R_{g, v}(f):= E^{\mathcal {A}}(v f*\delta _g)$ on $L^p(\mathcal {B})$, $1\le p<\infty$, where $\delta _g$ is the unit point mass measure at $g\in G$ and $E^{\mathcal {A}}$ is the conditional expectation operator associated with the $\sigma$-subalgebra $\mathcal {A}$. For an aperiodic element $g\in G$, we give the necessary and sufficient conditions on which $R_{g, v}$ is subspace-hypercyclic for $L^p(\mathcal {A})$ and $L^p(\mathcal {A}_D)$. The subspace-mixing concept for $R_{g, v}$ is also characterized. Furthermore, the subspace-hypercyclicity of the adjoint of $R_{g, v}$ with respect to $L^2(\mathcal {A}g^{-1})$ and other some specific subspaces is studied. Finally, some examples are then given to illustrate the obtained results.