Some relationships with subnormal operators and existence of hyperinvariant subspaces

If $T$ is a polynomially bounded operator, $\mathcal M$ is an invariant subspace of $T$, $T|_{\mathcal M}$ is a unilateral shift and $T^*|_{\mathcal M^\perp}$ is subnormal, then $T$ has a nontrivial hyperinvariant subspace. If an operator $T$ is intertwined from both sides with two operators, one of which is hyponormal and other is the adjoint to hyponormal, then $T$ has a nontrivial hyperinvariant subspace. The existence of nontrivial hyperinvariant subspaces for subnormal operators themselves is not studied here.