On the third problem of Halmos on Banach spaces

TL;DR

This paper investigates the intransitivity of invertible operators on Banach spaces, providing conditions under which the inverse operator shares intransitivity, and explores the existence of invariant subspaces in specific function spaces.

Contribution

It offers an affirmative answer to Halmos's third problem under spectral conditions and characterizes when certain function spaces lack non-trivial invariant subspaces.

Findings

Intransitivity of an invertible operator can be inferred from spectral properties.

Operators with Dunford's Property (C) and specific spectral conditions are intransitive.

Existence of non-trivial invariant subspaces in $L_1$ and $C(K)$ spaces depends on measure and compactness conditions.

Abstract

Assume that $X$ is a complex separable infinite dimensional Banach space and $\mathcal{B}(X)$ denotes the Banach algebra of all bounded linear operators from $X$ to itself. In 1970, P.R. Halmos raised ten open problems in Hilbert spaces. The third one is the following: If an intransitive operator $T$ has an inverse, is its inverse also intransitive? This question is closely related to the invariant subspace problem. Ever since Enflo's celebrated counterexample on $\ell_1$ answered the invariant subspace problem in negative, the Banach space setting of the third question of Halmos has become more interesting. In this paper, we give an affirmative answer to this problem under certain spectral conditions. As an application, we show that for an invertible operator $T$ with Dunford's Property ($C$), if $T^{-1}$ is intransitive and there exists a connected component $\Omega$ of $int\sigma(T^{-1})^\land$ which is off the origin such that $\Omega\cap\rho_F(T^{-1})\neq \emptyset$, then $T$ is also intransitive. In the end of the paper, we show that a sufficient and necessary condition for that there exists a bounded linear operator without non-trivial invariant subspaces on the infinite dimensional space $L_1(\Omega,\sum,\mu)$ (resp., $C(K)$, the space of bounded continuous functions on a complete metric space $K$) is that $(\Omega,\sum,\mu)$ is $\sigma$-finite (resp., $K$ is compact).