Some sufficient conditions for existence of hyperinvariant subspaces for operators intertwined with unitaries

TL;DR

This paper provides sufficient conditions for the existence of hyperinvariant subspaces in certain bounded operators, using singular inner functions and spectral analysis, with applications to quasianalytic contractions.

Contribution

It introduces new criteria for hyperinvariant subspaces for power bounded and polynomially bounded operators, expanding understanding of their structure and spectral properties.

Findings

Hyperinvariant subspaces can be constructed as closures of ranges of specific functions of the operator.

Conditions involve singular inner functions and spectral set properties.

An example of a quasianalytic contraction with partial spectral set characterization is provided.

Abstract

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of $\varphi(T)$. Here $\varphi$ is a singular inner function, if $T$ is polynomially bounded, or $\varphi$ is an analytic in the unit disc function with absolutely summable Taylor coefficients and singular inner part, if $T$ is supposed to be power bounded only. Also, an example of a quasianalytic contraction $T$ is given. The quasianalytic spectral set of $T$ is not the whole unit circle $\mathbb T$, while $\sigma(T)=\mathbb T$. Proofs are based on results by Esterle, Kellay, Borichev and Volberg.