On the structure of invariant subspaces for the shift operator on Bergman spaces

It is well known that the structure of nontrival invariant subspaces for the shift operator on the Bergman space is extremely complicated and very little is known about their specific structures, and that a complete description of the structure seems unlikely. In this paper, we find that any invariant subspace $M(\mathrm{\ne }\{0\})$ for the shift operator $M_z$ on the Bergman space $A_{\alpha }^{{}^p}(\text{D})(1\le p<\mathrm{\infty },-1<\alpha <\mathrm{\infty })$ contains a nonempty subset that lies in $A_{\alpha }^{{}^p}(\text{D})\mathrm{\backslash }{H}^{{}^p}(\text{D})$. To a certain extent, this result describes the specific structure of every nonzero invariant subspace for the shift operator $M_z$ on $A_{\alpha }^{{}^p}(\text{D})$.