Similarity of C1: Operators and the Hyperinvariant Subspace Problem

In the present paper, we 0rst show that the existence of the solutions of the operator equation S
XT � X is related to the similarity
of operators of class C1. , and then we give a su6cient condition for the existence of nontrivial hyperinvariant subspaces. +ese
subspaces are the closure of ran +(T) for some singular inner functions +. As an application, we prove that every C10-quasinormal
operator and C10-centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.