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

Abstract

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 $(\lambda -T{)}^{-1}\varphi$ of the operator $T$ has an analytic continuation from the lower half plane to the upper half plane as an $X^{\prime }$-valued holomorphic function for any $\varphi \in X$, even when $T$ has a continuous spectrum on $\mathbf{R}$, where $X^{\prime }$ is a dual space of $X$. The rigged Hilbert space consists of three spaces $X\subset \mathcal{H}\subset X^{\prime }$. A generalized eigenvalue and a generalized eigenfunction in $X^{\prime }$ are defined by using the analytic continuation of the resolvent as an operator from $X$ into $X^{\prime }$. 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.