On a class of $C_{\cdot0}$-contractions: hyperinvariant subspaces and intertwining operators

A class of $C_{\cdot0}$-contractions that is a generalization of the class of $C_{\cdot0}$-contractions with finite defect indices is considered. The results of [2,3] for $C_{\cdot0}$-contractions with finite defect indices are generalized: the lattices of hyperinvariant subspaces of such contraction $T$ is isomorphic to that of its Jordan model and is generated by subspaces of the form $\operatorname{Ker}\varphi(T)$ and $\operatorname{Ran}\varphi(T)$, where $\varphi\in H^\infty$. The form of the inverse to an isomorphism of the invariant subspace lattices given by an intertwining quasiaffinity is also studied. Next, for $C_{\cdot0}$-contractions in question, the quantity disc related to the lattice of invariant subspaces is computed.