# Invertibility of functions of operators and existence of hyperinvariant subspaces

## TL;DR

This paper investigates conditions under which certain operators possess nontrivial hyperinvariant subspaces, focusing on invertibility of functions of operators involving singular inner functions.

## Contribution

It establishes new criteria linking invertibility of functions of operators to the existence of hyperinvariant subspaces for polynomially bounded operators.

## Key findings

- Invertibility of $ heta(T)$ implies hyperinvariant subspaces under certain conditions
- Provides conditions involving singular inner functions and polynomially bounded operators
- Advances understanding of the structure of operators with hyperinvariant subspaces

## Abstract

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.