Backward shift invariant subspaces in the bidisc
Suppose that \(T_{\varphi}\) is a Toeplitz operator with symbol \(\varphi\) on the Hardy space \(H^{2}\) on the bidisc \(T^{2}\). Let \(N\) be a backward shift invariant subspace of \(H^{2}(T^{2})\), that is, \(N\) is invariant under \(T_{z}^{\ast}\) and \(T_{w}^{\ast}\), where \(z\) and \(w\) are the coordinate functions of the point of \(T^{2}\). Let \(P\) be the orthogonal projection from \(H^{2}(T^{2})\) onto \(N\). For \(\varphi\in H^{\infty}(T^{2})\), let \(S_{\varphi}=PT_{\varphi}| _{N}\). In the paper under review, the authors give a characterization of those backward shift invariant subspaces which satisfy the property \(S_{z}S_{w}^{\ast}=S_{w}^{\ast}S_{z}\). [For the sequel to this paper, see \textit{K. Izuchi, T. Nakazi} and \textit{M. Seto}, J. Oper. Theory 51, 361--376 (2004; Zbl 1055.47009), reviewed below.]
scientific article; zbMATH DE number 2092066