Multiplication by a finite Blaschke product on weighted Bergman spaces: commutant and reducing subspaces

TL;DR

This paper characterizes the commutant and reducing subspaces of multiplication operators induced by finite Blaschke products on weighted Bergman spaces, extending previous results and applying to other function spaces like Hardy spaces.

Contribution

It provides a new characterization of the commutant of Toeplitz operators with finite Blaschke products on weighted Bergman spaces, extending prior work and including Hardy spaces.

Findings

Characterization of the commutant of $T_B$ for finite Blaschke products

Extension of results to Hardy spaces $H^p$ for $1<p< olinebreak ext{infinity}$

Analysis of reducing subspaces of $T_B$ in weighted Bergman spaces

Abstract

We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.