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

Abstract

We provide a characterization of the commutant of analytic Toeplitz operators TB induced by finite Blaschke products B acting on weighted Bergman spaces which, as a particular instance, yields the case B(z)=zn on the Bergman space solved recently by Abkar, Cao and Zhu [1]. 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 Hp for 1<p<∞. Finally, we apply this approach to study the reducing subspaces of TB in weighted Bergman spaces.