The Beurling-type theorem for powers of weighted Bergman shifts

Let B_α denote the shift operator on the weighted Bergman space A²_α(D). We prove that, for every integer k≥1, the Beurling-type theorem holds for B_αᵏ on A²_α(D) if and only if -1<α≤1. We also show that, for every $α>-1$, there exists a finite Blaschke product B such that MB admits a cyclic invariant subspace but fails the wandering subspace property.