Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta$ be a singular inner function. It is shown that if $\theta(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial hyperinvariant subspaces.
Some sufficient conditions for existence of hyperinvariant subspaces for operators intertwined with unitaries
On existence of hyperinvariant subspaces for quasinilpotent operators with a nonsymmetry in the growth of the resolvent
Invertibility, Fredholmness and kernels of dual truncated Toeplitz operators
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