Wandering subspace property of the shift operator on a class of invariant subspaces of the weighted Bergman space 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)circle minus K-0. In this paper, by lifting the shift operator 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 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 B-2 on some H-a type submodules of L-a(2)(dA(2)).