The wandering subspace property and Shimorin’s condition of shift operator on the weighted Bergman spaces

In the present paper, we first study the wandering subspace property of the shift operator on the $$I_{a}$$ type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{n})(n=0,2)$$ via the spectrum of some Toeplitz operators on the Hardy space $$H^{2}$$ . Second, we give examples to show that Shimorin’s condition for the shift operator fails on the $$I_{a}$$ type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{\alpha })(\alpha >0)$$ .