Approximate synthesizability of coinvariant subspaces of isometric shift operators of finite multiplicity (Q1071281)

Let E be a separable complex Hilbert space, \(S_ E\) be an isometric shift of multiplicity dim E in the Hardy space \(H^ 2(E)\) and \(S^*_ E\) be its adjoint; denote by Lat \(S^*_ E\) the lattice of \(S^*_ E\)-invariant subspaces of E. The following theorem is proved: If dim E\(<\infty\), then for every \({\mathfrak M}\in Lat S^*_ E\) there is a sequence \(\{\) \({\mathfrak M}_ n\}\) of finite dimensional \(S^*_ E\)-invariant subspaces of E such that \[ {\mathfrak M}=\lim {\mathfrak M}_ n\quad and\quad {\mathfrak M}^{\perp}=\lim {\mathfrak M}_ n^{\perp}. \] Some auxiliary results are proved for E of an arbitrary dimension.