Hyperinvariant subspaces for operators intertwined with weighted shift operators

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