  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
$M(\neq \{0\})$ for the shift operator $M_z$ on the Bergman space
$A^{^p}_\alpha(\D)~(1\leq p< \infty,~~-1<\alpha<\infty)$ contains a nonempty
subset that lies in $A^{^p}_\alpha(\D)\backslash H^{^p}(\D)$. To a certain
extent, this result describes the specific structure of every nonzero invariant
subspace for the shift operator $M_z$ on $A^{^p}_\alpha(\D)$.
