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

A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions