The Beurling-type theorem for powers of weighted Bergman shifts

Let $\mathcal{B}_α$ denote the shift operator on the weighted Bergman space $A^2_α(\mathbb{D})$. We prove that, for every integer $k\geq1$, the Beurling-type theorem holds for $\mathcal{B}_α^k$ on $A^2_α(\mathbb{D})$ if and only if $-1<α\leq1$. We also show that, for every $α>-1$, there exists a finite Blaschke product $B$ such that $M_{B}$ admits a cyclic invariant subspace but fails the wandering subspace property.