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

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. 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.