  Let $T$ be a bounded linear operator on a complex separable infinite
dimensional Hilbert space $\mathcal{H}$. $T$ is called intransitive if it
leaves invariant spaces other than 0 or the whole space $\mathcal{H}$;
otherwise it is transitive. In 1970, P. R. Halmos raised ten open problems on
operator theory. In the past more than 50 years, nine of Halmos' ten problems
were answered, but only the third one has made little progress. The third
problem of Halmos is the following: if an intransitive operator has an inverse,
is its inverse also intransitive? In this paper, we establish a set of
theoretical systems with the help of Cowen-Douglas operators and spectral
analysis. We give an affirmative answer to this problem under certain spectral
conditions, which make essential progress in the research of Halmos' third
problem. As the first application, we show that for an invertible hyponormal
operator $T$, if $T^{-1}$ is intransitive and int$\sigma(T^{-1})^{\land}$ is
not connected, then $T$ is also intransitive. As the second application, we
show that if $T^{-1}$ has a proper strictly cyclic invariant subspace and there
exists a bounded open set $\Omega$ which is a connected component of
$\rho(T^{-1})$ such that $\Omega\cap \mathcal{U}_0=\emptyset$, where
$\mathcal{U}_0$ is the connected component of $int(\sigma(T^{-1})^\land)$
containing zero point, then $T$ is intransitive.
