2022/03/28 by Lixin Cheng, Cheng, Lixin, Junsheng Fang +3
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2203.14670
openalex publication_date 2022/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Assume that X is a complex separable infinite dimensional Banach space and 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 ℓ1 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 Ω of intσ(T-1)^∧ which is off the origin such that Ω∩ρF(T-1)≠ ∅, 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 L1(Ω,∑,μ) (resp., C(K), the space of bounded continuous functions on a complete metric space K) is that (Ω,∑,μ) is σ-finite (resp., K is compact).