# Cowen-Douglas operators and the third of Halmos' ten problems

## TL;DR

This paper investigates Halmos' third problem on intransitive operators with inverses, providing affirmative answers under spectral conditions using Cowen-Douglas operators and spectral analysis, advancing understanding in operator theory.

## Contribution

The paper offers new theoretical results affirming that intransitivity is preserved under inversion for certain classes of operators, addressing a long-standing open problem in operator theory.

## Key findings

- Affirmative answer under spectral conditions for intransitivity preservation.
- Inversion of hyponormal operators retains intransitivity if spectral conditions are met.
- Intransitivity is preserved for operators with specific spectral and invariant subspace properties.

## Abstract

Let $T$ be a bounded linear operator on a complex separable infinite dimensional Hilbert space $\mathcal{H}$. $T$ is called intransitive if it leaves invariant spaces other than 0 or the whole space $\mathcal{H}$; otherwise it is transitive. In 1970, P. R. Halmos raised ten open problems on operator theory. In the past more than 50 years, nine of Halmos' ten problems were answered, but only the third one has made little progress. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? In this paper, we establish a set of theoretical systems with the help of Cowen-Douglas operators and spectral analysis. We give an affirmative answer to this problem under certain spectral conditions, which make essential progress in the research of Halmos' third problem. As the first application, we show that for an invertible hyponormal operator $T$, if $T^{-1}$ is intransitive and int$\sigma(T^{-1})^{\land}$ is not connected, then $T$ is also intransitive. As the second application, we show that if $T^{-1}$ has a proper strictly cyclic invariant subspace and there exists a bounded open set $\Omega$ which is a connected component of $\rho(T^{-1})$ such that $\Omega\cap \mathcal{U}_0=\emptyset$, where $\mathcal{U}_0$ is the connected component of $int(\sigma(T^{-1})^\land)$ containing zero point, then $T$ is intransitive.