2011/07/29 by Chiba, Hayato
#47A10 #47A75 #47D06 #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1107.5858
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.