Composition operators acting between some weighted Möbius invariant spaces

Abstract

‎In this paper we investigate conditions under which a holomorphic‎ ‎self-map of the unit disk induces a composition operator $C_\phi$‎ ‎with closed range on the weighted Bloch space $\mathcal{B}_{\log}$.‎ ‎Also‎, ‎we introduce a new class of functions the so called‎ ‎$F_{\log}(p,q,s)$ spaces‎. ‎Necessary and sufficient conditions are‎ ‎given for a composition operator $C_\phi$ to be bounded and compact‎ ‎from ${\mathcal{B}_{\log}}$ to $F_{\log}(p,q,s)$. Moreover‎, ‎necessary and sufficient conditions for $C_{\phi}$ from the‎ ‎Dirichlet space $\mathcal{D}$ to the spaces $F_{\log}(p,q,s)$ to be‎ ‎compact are given in terms of the map $\phi$‎.