TY - JOUR

T1 - A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions

AU - Chiba, Hayato

N1 - Funding Information:
This work was supported by Grant-in-Aid for Young Scientists (B), No. 22740069 from MEXT Japan.
Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2015/3/9

Y1 - 2015/3/9

N2 - 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.

AB - 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.

KW - Gelfand triplet

KW - Generalized eigenvalue

KW - Generalized function

KW - Resonance pole

KW - Rigged Hilbert space

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U2 - 10.1016/j.aim.2015.01.001

DO - 10.1016/j.aim.2015.01.001

M3 - Article

AN - SCOPUS:84921327052

VL - 273

SP - 324

EP - 379

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -