Wandering subspace property of the shift operator on a class of invariant subspaces of the weighted Bergman space $$L_{a}^{2}(dA_{2})$$La2(dA2)

Let $$H^{2}(D^{2})$$ be the Hardy space over the bidisk $$D^{2}$$, and let $$K_{0}=[(z-w)^{2}]$$ be the submodule generated by $$(z-w)^{2}$$. The related quotient module is $$N_{0}=H^{2}(D^{2})\ominus K_{0}$$. In this paper, by lifting the shift operator $${\mathcal {B}}_{2}$$ on the weighted Bergman space $$L_{a}^{2}(dA_{2})$$ as the compression of an isometry on a closed subspace of $$N_{0}$$, we prove that the shift operator $${\mathcal {B}}_{2}$$ possesses wandering subspace property on the $$H_{a}$$ type submodules of $$L_{a}^{2}(dA_{2})$$. Also we show that Shimorin’s condition fails for $${\mathcal {B}}_{2}$$ on some $$H_{a}$$ type submodules of $$L_{a}^{2}(dA_{2})$$.