The Beurling-type theorem in the Bergman space $A^2_\alpha(D)$ for any $-1<\alpha<+\infty$

TL;DR

This paper proves a Beurling-type theorem for Bergman spaces with a wide range of weights, demonstrating that all invariant subspaces are generated by their orthogonal complements under the shift operator.

Contribution

The authors introduce a new method to establish the Beurling-type theorem in Bergman spaces for all weights in the range -1<α<+∞, extending previous results.

Findings

Proved the Beurling-type theorem for all α in -1<α<+∞

Showed invariant subspaces are generated by their orthogonal complements

Extended the theorem to a broader class of weighted Bergman spaces

Abstract

In this paper, we use a new method to solve a long-standing problem. More specifically, we show that the Beurling-type theorem holds in the Bergman space $A^2_\alpha(D)$ for any $-1<\alpha < +\infty$. That is, every invariant subspace $H$ for the shift operator $S$ on $A^2_\alpha(D)$ $(-1<\alpha < +\infty)$ has the property $H=[H\ominus zH]_{S,A^2_\alpha\left(D\right)}$.