Non-stable subnormal contractions have nontrivial hyperinvariant subspaces

TL;DR

The paper proves that non-stable pure subnormal contractions on Hilbert spaces have nontrivial hyperinvariant subspaces, extending understanding of their structure using singular inner functions.

Contribution

It establishes the existence of nontrivial hyperinvariant subspaces for a class of non-stable subnormal contractions, based on advanced functional analysis results.

Findings

Non-stable pure subnormal contractions have nontrivial hyperinvariant subspaces.

Existence of singular inner functions related to these contractions.

Examples of stable subnormal contractions with dense range for all nonzero bounded analytic functions.

Abstract

A contraction $T$ on a (complex, separable) Hilbert space is stable, or of class $C_{0\cdot}$, if $T^n\to 0$ in the strong operator topology. It is proved that for a non-stable pure subnormal contraction $T$ there exists a singular inner function $\theta$ such that the range of $\theta(T)$ is not dense. Consequently, $T$ has nontrivial hyperinvariant subspaces. The proof is based on results by Esterle and K\'erchy. Examples of stable subnormal contractions are given for which the range of $\varphi(T)$ is dense for every $\varphi\in H^\infty$ ($\varphi\not\equiv 0$).