The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces

Abstract

For every $\alpha >1$ we construct a finite set $A\subset \mathbb{D}$ such that the zero-based invariant subspace $I_A$ fails the wandering subspace property. Consequently, the Beurling-type theorem holds on the weighted Bergman space $A_{\alpha }^2$ if and only if $-1<\alpha \le 1$, confirming a conjecture of Shimorin. As a further application, we extend the prescribed-curvature construction of Hedenmalm and Perdomo to all $\alpha >1$, thereby replacing the constant ${\alpha }_0\approx 1.04$ in their result by $1$.

34 pages