  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).
