Abstract
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 \(L_{a}^{2}(dA_{2})\).
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Acknowledgements
The authors are very grateful to professor Rongwei Yang for suggesting this line of research, and for his ongoing encouragement and support. Chong Zhao of Shandong University provided a good idea in the proof of the last theorem in this paper. We are very grateful to him here. We also thank the referees for their excellent suggestions and precious time. This research is supported by NSFC (grant no., 11971087 and 11431011).
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Communicated by Raul Curto.
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Wu, C., Yu, T. Wandering subspace property of the shift operator on a class of invariant subspaces of the weighted Bergman space \(L_{a}^{2}(dA_{2})\). Banach J. Math. Anal. 14, 784–820 (2020). https://doi.org/10.1007/s43037-019-00039-9
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DOI: https://doi.org/10.1007/s43037-019-00039-9