Dynamics on weighted solid Banach function spaces
Stefan Ivkovic
TL;DR
The paper generalizes the study of dynamics of weighted translations to weighted solid Banach function spaces by introducing weighted solid spaces $\mathcal{F}_{\eta}$ and weighted composition operators $T_{\alpha,w}$ on them. It provides a precise, checkable characterization: $T_{\alpha,w}$ is topologically transitive on $\mathcal{F}_{\eta}$ if and only if, for every compact $K$, there exist sequences $\{n_k\}$ and $\{E_k\}$ with $\|\,\chi_{K\setminus E_k}\|_{\mathcal{F}}\to 0$ and weighted-product limits tending to $0$ both for $\alpha^{n_k}$ and $\alpha^{-n_k}$; this criterion extends to disjoint families of weighted translations and to multiple operators via conditions (4.1)-(4.4). The results apply to weighted Morrey spaces and provide a multi-operator generalization, unifying the dynamics of weighted translations in a broad class of spaces. Overall, the work advances the understanding of hypercyclic-like dynamics in weighted solid Banach spaces and offers practical criteria for verifying disjoint transitivity and related properties.
Abstract
The dynamics of weighted translation operators on Lebesgue spaces, Orlicz spaces, and in general on solid Banach function spaces have been studied in numerous papers. Recently, the dynamics of weighted translations on weighted Orlicz spaces have also been studied by Chen and others. The main idea of this paper is to obtain a generalization of these results to the case of general weighted solid Banach function spaces. More precisely, in this paper, we characterize disjoint topologically transitive and disjoint supercyclic weighted composition operators on weighted solid Banach function spaces. This approach has applications in the dynamics of weighted translations on weighted Morrey spaces.