Rank-one cross commutators on backward shift invariant subspaces on the bidisk (Q1034295)

Let \(L^2=L^2(\Gamma^2)\) be the Lebesgue space and \(H^2=H^2(\Gamma^2)\) be the Hardy space over \[ \Gamma^2= \{(z,w):z,w\in {\mathbb C},\;| z|=1,\;| w|=1\}. \] Let \(P\) be the orthogonal projection from \(L^2\) onto \(H^2\). For \(\psi\in H^\infty (\Gamma^2)\), the Toeplitz operator \(T_\psi\) is defined in the standard manner: \(T_\psi=P\psi I\). A subspace \(M \subset H^2\) is called invariant if \(zM \subset M\) and \(wM \subset M\). It is supposed that \(M\) is nontrivial. For such subspace \(M\), denote \(N=H^2\ominus M\). For each \(\psi\in L^\infty(\Gamma^2)\), define the operator \(S_\psi\) on \(N\) by the equality \( S_\psi=P_N T_\psi|_{N} \), where \(P_N\) is the orthogonal projection from \(H^2\) onto~\(N\). The main result of the paper gives the full description of the subspaces \(M\) for which the commutator \([S_z,S_w^*]\) has rank one.