On Halmos’ third problem on Banach spaces

In 1970, Halmos posed ten open problems in Hilbert spaces. The third problem asks: If an intransitive operator T has an inverse, is its inverse also intransitive? This question is closely related to the invariant subspace problem. Since Enflo’s celebrated counterexample on ℓ₁ provided a negative answer to the invariant subspace problem, the Banach space version of Halmos’ third problem has attracted increasing interest. In this paper, we provide 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 int σ(T⁻¹)^⋀ off the origin and satisfies Ω ⋂ ρ_F(T⁻¹) ≠ ∅, then T is also intransitive. Finally, we show that a necessary and sufficient condition for the existence of a bounded linear operator without nontrivial invariant subspaces on the infinite-dimensional space L₁(Ω, Σ, μ) (resp. C(K), the space of bounded continuous functions on a complete metric space K) is that (Ω, Σ, μ)is σ-finite (resp. K is compact).