One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk. I
Let \(H^{2}=H({\mathbb D}^{2})\) be the Hardy space over the bidisk \({\mathbb D}^{2}\), points of \({\mathbb D}^{2}\) are denoted as \((z,w)\). A~nonzero space \(M_{1}\subset H^{2}\) is called invariant if \(zM_{1}\subset M_{1}\) and \(wM_{1}\subset M_{1}\). For such a subspace \(M_{1}\), denote \(\Omega(M_{1})=M_{1}\ominus (zM_{1}+wM_{1})\), \(N_{1}=H^{2}\ominus M_{1}\), \( \widetilde{\Omega}(N_{1}) = \{ f\in N_{1}: zf,\, wf\in M_{1} \} \). Then \(\Omega(M_{1})\) and \(\widetilde{\Omega}(N_{1})\) are subspaces of \(H^{2}\) and \(\Omega(M_{1})\neq \{ 0\}\). It has been known that there exists such nonzero \(f_{0}\in M_{1}\) that the space \(M_{2} =M_{1}\ominus \operatorname{span}(f_{0})\) is an invariant subspace. The aim of the paper under review is to describe the structure of the subspaces \(\Omega(M_{2})\) and \(\widetilde{\Omega}(N_{2})\) in terms of \(\Omega(M_{1})\), \(\widetilde{\Omega}(N_{1})\), and \(f_{0}\). We describe one of the obtained results. Suppose that \(f_{0}\not\in M_{1}\ominus zM_{1}\) and \(f_{0}\in M_{1}\ominus wM_{1}\). Then the following properties hold: (1) if \(f_{0}\perp \Omega(M_{1})\), then \(\Omega(M_{2})=\Omega(M_{1})\); (2) if \(f_{0}\not\perp \Omega(M_{1})\), then there exists a nonzero function \(h_{0}\in (M_{1}\ominus wM_{1})\ominus\operatorname{span}(f_{0})\) such that \( \Omega(M_{2}) = (\Omega(M_{1})\ominus\operatorname{span}(P_{\Omega(M_{1})}f_{0})))\oplus\operatorname{span}(h_{0}), \) where \(P_{K}\) as usual denotes the orthogonal projection on the subspace \(K\subset H^{2}\).
scientific article; zbMATH DE number 6881239