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 $H=[Hominus zH]_{S,A^2_alphaleft(Dright)}$.