One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk. II (Q1635286)

The paper continues the investigation of Part I [the authors, ibid. 28, No. 1, 1--29 (2017; Zbl 1470.47011)]. Let \(H^{2}=H^{2}(\mathbb D^{2})\) be the Hardy space over the bidisk \({\mathbb D}^{2}\) with variable \((z,w)\). Let \(T_{z}\) (\(T_{w}\)) be the multiplication operator on \(H^{2}\) by \(z\) (\(w\)). A~nonzero subspace \(M_{1}\subset H^{2}\) is said to be invariant if \(T_{z}\subset M_{1}\) and \(T_{w}\subset M_{1}\). Denote \(N_{1}=H^{2}\ominus M_{1}\). There are defined operators \(R^{M_{1}}_{z}=T_{z}|_{M_{1}}\), \(S_{z}^{N_{1}}=P_{N_{1}}T_{z}|_{N_{1}}\) and corresponding operators \(R^{M_{1}}_{w}\), \(S_{w}^{N_{1}}\). Here, \(P_{N_{1}}\) is the orthogonal projection on the subspace \(N_{1}\). It is known that, for an invariant subspace \(M_{1}\), there exists a nonzero \(f_{0}\in M_{1}\) such that \(M_{2}=M_{1}\ominus \operatorname{span}(f_{0})\) is also an invariant subspace. The authors investigate relations between the ranks of commutators \([R_{w}^{\ast},R_{z}]\) and \([S_{w},S_{z}^{\ast}]\) connected with the spaces \(M_{1}\) and \(M_{2}\). For the latter case, the following estimate is obtained: \( \operatorname{rank}[S_{w}^{N_{1}},S_{z}^{N_{1}\ast}] - 1 \leq \operatorname{rank}[S_{w}^{N_{2}},S_{z}^{N_{2}\ast}] \leq \operatorname{rank}[S_{w}^{N_{1}},S_{z}^{N_{1}\ast}] + 3 \).