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

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 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 H such that the resolvent (λ-T)^-1ϕ 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 ϕ∈X, even when T has a continuous spectrum on R, where X^' is a dual space of X. The rigged Hilbert space consists of three spaces X⊂H⊂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.