Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

Abstract

Let M be a shift invariant subspace in the vector-valued Hardy space \({H_{E}^{2}(\mathbb{D})}\). The Beurling–Lax–Halmos theorem says that M can be completely characterized by \({\mathcal{B}(E)}\)-valued inner function \({\Theta}\). When \({E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}\) is the Hardy space on the bidisk \({H^{2}(\mathbb{D}^2)}\). Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function \({\Theta(z)}\) for two well-known invariant subspaces in \({H^{2}(\mathbb{D}^{2})}\). This paper generalizes the \({\Theta(z)}\) by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of \({M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}\) when M is an invariant subspace in \({H^{2}(\mathbb{D}^{2})}\). Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.