Ranks of cross commutators on backward shift invariant subspaces over the bidisk (Q983500)

Let \(L^{2}=L^{2}(\Gamma^{2})\) be the Lebesgue space and \(H^{2}=H^{2}(\Gamma^{2})\) be the Hardy space over \[ \Gamma^{2}= \{ (z,w):z,w\in {\mathbb C},\;| z|=1,\;| w|=1 \}. \] A subspace \(M \subset H^{2}\) is called invariant if \(zM \subset M\) and \(wM \subset M\). Suppose that \(M\) is nontrivial. For a function \(\psi \in L^{\infty}(\Gamma^{2})\), denote by \(R_{\psi}\) the operator acting on the space \(M\) by the formula \(R_{\psi}f=P_{M}(\psi f)\), \(f\in M\), where \(P_{M}\) is the orthonormal projection in \(L^{2}\) on the subspace \(M\). Denote \(N=H^{2}\ominus M\) and, for the same function \(\psi\), define the operator \(S_{\psi}\) on \(N\) by the equality \(S_{\psi}f=P_{N}(\psi f)\), \(f\in N\). In [Acta Math.\ Sin., Engl.\ Ser.\ 25, No.\,5, 693--714 (2009; Zbl 1193.47011)], the authors made the following conjecture: \[ \text{rank}[R_{z},R_{w}^{\ast}]-1 \leq \text{rank} [S_{z},S_{w}^{\ast}] \leq \text{rank}[R_{z},R_{w}^{\ast}]+1, \] where \([\cdot,\cdot]\) denotes the commutator of operators. In the paper under review, it is proved that this inequality is valid for subspaces \(M\) of some special class.