  A spectral theory of linear operators on rigged Hilbert spaces (Gelfand
triplets) is developed under the assumptions that a linear operator $T$ on a
Hilbert space $\mathcal{H}$ is a perturbation of a selfadjoint operator, and
the spectral measure of the selfadjoint operator has an analytic continuation
near the real axis in some sense. It is shown that there exists a dense
subspace $X$ of $\mathcal{H}$ such that the resolvent $(\lambda -T)^{-1}\phi$
of the operator $T$ has an analytic continuation from the lower half plane to
the upper half plane as an $X'$-valued holomorphic function for any $\phi \in
X$, even when $T$ has a continuous spectrum on $\mathbf{R}$, where $X'$ is a
dual space of $X$. The rigged Hilbert space consists of three spaces $X \subset
\mathcal{H} \subset X'$. A generalized eigenvalue and a generalized
eigenfunction in $X'$ are defined by using the analytic continuation of the
resolvent as an operator from $X$ into $X'$. Other basic tools of the usual
spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup
are also studied in terms of a rigged Hilbert space. They prove to have the
same properties as those of the usual spectral theory. The results are applied
to estimate asymptotic behavior of solutions of evolution equations.
