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

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 $A^2_\alpha(D)$ 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 $H=[H\ominus zH]{S,A^2\alpha\left(D\right)}$.