A characterization of topologically multiply recurrent weighted translation operators on solid Banach function spaces

In this paper, we present an equivalent condition for a sequence \((T_{\alpha ,w}^{m_l})_{l=1}^\infty\) ${(T_{\alpha ,w}^{m_l})}_{l=1}^{\infty }$ of weighted translations on \({\mathcal {F}}\) $\mathcal{F}$ to be topologically multiply recurrent, where \({\mathcal {F}}\) $\mathcal{F}$ is a solid Banach function space on \(\Omega\) $\mathrm{\Omega }$ , \(\alpha :\Omega \longrightarrow \Omega\) $\alpha :\mathrm{\Omega }⟶\mathrm{\Omega }$ is an aperiodic measure space isomorphism, w is a weight on \(\Omega\) $\mathrm{\Omega }$ , and \((m_l)_{l\in {\mathbb {N}}}\subseteq {\mathbb {N}}\) ${(m_l)}_{l\in \mathbb{N}}\subseteq \mathbb{N}$ is unbounded. The given results are generalizations of the previous ones and also, we give several applications in the situations of group action and some closed subspaces of Morrey spaces.