Description of hyperinvariant subspaces of a contraction in terms of its characteristic function

If $T$ is a completely nonunitary contraction on a Hilbert space and $L$ is its invariant subspace corresponding to a regular factorizations of its characteristic function $\Theta=\Theta'\Theta''$, then $L$ is hyperinvariant if and only if the following two conditions are fulfilled:

\item[$1\circ)$] $\operatorname{supp}\Delta'_*\cap\operatorname{supp}\Delta''$ is of Lebesgue measure zero;
\item[$2\circ)$] for every pair $A\in H^{\infty}(E\to E)$, $A_*\in H^{\infty}({E_*}\to{E_*})$ intertwinned by $\Theta$, i.e., such that $\Theta A=A_*\Theta$, there exists a function $A_F\in H^{\infty}(F\to F)$ intertwinned by $\Theta'$ with $A$ and by $\Theta'$ with $A$ and by $\Theta''$ with $A_*$, i.e., $\Theta'A=A_F\Theta'$, $\Theta'' A_F=A_*\Theta''$.