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

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