Hyperinvariant subspaces for operators intertwined with weighted shift operators

Suppose that \(T\) is an absolutely continuous polynomially bounded operator, \(S_{\omega}\) is a bilateral weighted shift, there exists a \(\phi\in \mathbb{H}^{\infty}\) such that \(\ker \phi(S_{\omega}^{*})\neq \{0\}\) and a nonzero operator \(X\) such that \(S^{(\infty)}_{\omega}X=XT\), where \(S^{(\infty)}_{\omega}\) is the infinite countable orthogonal sum of copies of \(S_{\omega}\). We prove that \(T\) has nontrivial hyperinvariant subspaces, that are the closures of \(\text{Ran} \psi(T)\) for some \(\psi \in \mathbb{H}^{\infty}\).