Lattices of invariant subspaces for a quasiaffine transform of a unilateral shift of finite multiplicity (Q1781761)

An interesting result of this paper is as follows: Assume that \(H\) and \(E\) are separable Hilbert spaces such that \(\dim E= k<\infty\). Denote by \(H^2(E)\) the \(H^2\)-functions from the unit circle \(\partial\mathbb{D}\) to \(E\). Let \(T: H\to H\) be a contraction, let \(S: H^2(E)\to H^2(E)\) be the unilateral shift, and let an operator \(X: H\to H^2(E)\) be such that \(\ker X=\{0\}\), \(\text{clos\,}XH= H^2(E)\), and \(XT= SX\). Then there exist families \(\{Y_i\}_{i\in I}\) of operators and \(\{\delta_i\}_{i\in I}\) of functions \(Y_i: H^2(E)\to H\), \(\delta_i: H^\infty\), such that \(TY_i= Y_iS\), \(XY_i= \delta_i(S)\), \(Y_iX= \delta_i(T)\) for any \(i\in I\), and the greatest common divisor of the inner parts of the functions \(\{\delta_i\}_{i\in I}\) equals \(I\).