Similarity of C1: Operators and the Hyperinvariant Subspace Problem

In the present paper, we first 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 sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of ranφT for some singular inner functions φ . As an application, we prove that every C10 -quasinormal operator and C10 -centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.