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 HH 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 HH such that the resolvent (λ−T)−1ϕ(λ−T)−1ϕ of the operator T   has an analytic continuation from the lower half plane to the upper half plane as an X′X′-valued holomorphic function for any ϕ∈Xϕ∈X, even when T has a continuous spectrum on R, where X′X′ is a dual space of X  . The rigged Hilbert space consists of three spaces X⊂H⊂X′X⊂H⊂X′. A generalized eigenvalue and a generalized eigenfunction in X′X′ are defined by using the analytic continuation of the resolvent as an operator from X   into X′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.