On the third problem of Halmos on Banach spaces

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).