Rank-one commutators on invariant subspaces of the Hardy space on the bidisk
Let \(H^2(\Gamma^2)\) be the Hardy space on the two-dimensional torus on \(\Gamma^2\). Let \(T_z\) and \(T_w\) be the multiplication operators on \(H^2(\Gamma^2)\) by the coordinate functions \(z\) and \(w\), respectively. Let \(M\) be an invariant subspace in \(H^2(\Gamma^2)\), that is, \(T_zM\subset M\) and \(T_w M\subset M\). For an invariant subspace \(M\) of \(H^2(\Gamma^2)\), put \(V_w= P_M T_z\) and \(V_w= P_M T_w\) on \(M\), where \(P_M\) is the orthogonal projection onto \(M\). Write \([V_w, V^*_z]= V_w V^*_z- V^*_z V_w\). The authors describe an invariant subspace \(M\) in \(H^2(\Gamma^2)\) such that \([V_m, V^*_z]\) is a nonzero selfadjoint operator and they show that \(\text{rank}[V_w, V^*_z]= 1\). Moreover, they study some examples of invariant subspaces in \(H^2(\Gamma^2)\) such that \(\text{rank}[V_w, V^*_z]<\infty\).
scientific article; zbMATH DE number 5016162
scientific article