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

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 $A_{\alpha }^2(D)$ for any $-1<\alpha <+\infty$. That is, every invariant subspace $H$ for the shift operator $S$ on $A_{\alpha }^2(D)$ $(-1<\alpha <+\infty )$ has the property $H=[H\ominus zH{]}_{S,A_{\alpha }^2\left(D\right)}$.