  Let $\mathcal{W}$ be the corresponding wandering subspace of an invariant
subspace of the Bergman shift. By identifying the Bergman space with
$H^2(\mathbb{D}^2)\ominus[z-w]$, a sufficient and necessary conditions of a
closed subspace of $H^2(\mathbb{D}^2)\ominus[z-w]$ to be a wandering subspace
of an invariant subspace is given also, and a functional charaterization and a
coefficient characterization for a function in a wandering subspace are given.
As a byproduct, we proved that for two invariant subspaces $\mathcal{M}$,
$\mathcal{N}$ with $\mathcal{M}\supsetneq\mathcal{N}$ and
$dim(\mathcal{N}\ominus B\mathcal{N})<\infty$ $dim(\mathcal{M}\ominus
B\mathcal{M})=\infty$, then there is an invariant subspace $\mathcal{L}$ such
that $\mathcal{M}\supsetneq\mathcal{L}\supsetneq\mathcal{N}$. Finally, we
define an operator from one wandering subspace to another, and get a
decomposition theorem for such an operator which is related to the universal
property of the Bergman shift.
