Hypercyclicity of Weighted Composition Operators on $L^p$-Spaces

Abstract

Let $(X, \Sigma , \mu )$ be a $\sigma$-finite measure space and $W=uC_{\varphi }$ be a weighted composition operator on $L^p(\Sigma )$ ($1\le p<\infty$), defined by $W:f\mapsto u.(f\circ \varphi )$, where $\varphi : X\rightarrow X$ is a measurable transformation and u is a weight function on X. In this paper, we study the hypercyclicity of W in terms of u, using the Radon–Nikodym derivatives and the conditional expectations. First, it is shown that if $\varphi$ is a periodic nonsingular transformation, then W cannot be hypercyclic. The necessary conditions for the hypercyclicity of W are then given in terms of the Radon–Nikodym derivatives provided that $\varphi$ is non-singular and finitely non-mixing. For the sufficient conditions, we also require that $\varphi$ is normal. The weakly mixing and topologically mixing concepts are also studied for W. Moreover, under some specific conditions, we establish the subspace hypercyclicity of the adjoint operator $W^*$ with respect to the Hilbert subspace $L^2(\mathcal {A})$. Finally, to illustrate the results, some examples are given.