On a class of unitary operators on weighted Bergman spaces

Abstract

In this paper we consider a class of weighted composition operators defined on the weighted Bergman spaces $L^{2}_a(dA_\alpha)$ where $\mathbb{D}$ is the open unit disk in $\mathbb{C}$ and $dA_\alpha(z)=(\alpha +1) (1-|z|^2)^\alpha dA(z),\; \alpha >-1$ and $dA(z)$ is the area measure on $\mathbb{D}$. We establish that a bounded linear operator $S$ from $L^{2}_a(dA_\alpha)$ into itself commutes with all the composition operators $C_a^{(\alpha)},\; a\in\mathbb{D}$, if and only if $B_\alpha S$ satisfies certain averaging condition. Here $B_\alpha S$ denotes the generalized Berezin transform of the bounded linear operator $S$ from $L^{2}_a(dA_\alpha)$ into itself. Applications of the result are also discussed. Further, we have shown that if $\mathcal{M}$ is a subspace of $L^\infty(\mathbb{D})$ and if for $\phi \in \mathcal{M},$ the Toeplitz operator $T_\phi^{(\alpha)}$ represents a multiplication operator on a closed subspace $\mathcal{S} \subset L^2_a(dA_\alpha),$ then $\phi$ is bounded analytic on $\mathbb{D}$. Similarly if $q\in L^\infty(\mathbb{D})$ and $B_n$ is a finite Blaschke product and $M_q^{(\alpha)}\left(Range \; C_{B_n}^{(\alpha)}\right)\subset L^2_a(dA_\alpha),$ then $q\in H^\infty(\mathbb{D}).$ Further, we have shown that if $\psi \in Aut(\mathbb{D})$ and $q\in L^2_a(dA_\alpha),$ then $\mathcal{N}= \left\lbrace q\in L^2_a(dA_\alpha): M_q^{(\alpha)}\left(Range\; C_\psi^{(\alpha)}\right) \subset L^2_a(dA_\alpha)\right\rbrace =H^\infty(\mathbb{D})$ if and only if $\psi$ is a finite Blaschke product.