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 $T_B$ 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 reducing subspaces of $T_{B}$ in the classical Bergman space. As a particular instance, we provide a direct proof of a theorem of Hu, Sun, Xu and Yu which states that every analytic Toeplitz operator $T_B$ induced by a finite Blachke product on the Bergman space is reducible and the restriction of $T_B$ on a reducing subspace is unitarily equivalent to the Bergman shift.