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

TL;DR

This paper develops a spectral theory for linear operators on rigged Hilbert spaces under analyticity conditions, extending classical spectral concepts to operators with continuous spectra and applying results to evolution equations.

Contribution

It introduces a spectral framework for operators on rigged Hilbert spaces with analytic spectral measures, including generalized eigenvalues and eigenfunctions, and extends classical spectral tools.

Findings

Existence of an analytic continuation of the resolvent in rigged Hilbert spaces.

Definition of generalized eigenvalues and eigenfunctions via analytic continuation.

Application to asymptotic analysis of evolution equations.

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 $(\lambda -T)^{-1}\phi$ 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.