Similarity of $C_1$: Operators and the Hyperinvariant Subspace Problem

In the present paper, we first show that the existence of the solutions of the operator equation $S^{\ast }\text{XT}=X$ is related to the similarity of operators of class $C_{1.}$, and then we give a sufficient condition for the existence of nontrivial hyperinvariant subspaces. These subspaces are the closure of $\text{ran}\phi \left(T\right)$ for some singular inner functions $\phi$. As an application, we prove that every $C_{10}$-quasinormal operator and $C_{10}$-centered operator, under suitable conditions, have nontrivial hyperinvariant subspaces.