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

Hayato Chiba

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)324-379
Number of pages56
JournalAdvances in Mathematics
Volume273
DOIs
Publication statusPublished - Mar 9 2015

Fingerprint

Spectral Theory
Analyticity
Linear Operator
Analytic Continuation
Hilbert space
Resolvent
Self-adjoint Operator
Half-plane
Generalized Eigenvalue
Spectral Measure
Continuous Spectrum
Dual space
Asymptotic Behavior of Solutions
Operator
Evolution Equation
Eigenfunctions
Analytic function
Semigroup
Subspace
Projection

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions. / Chiba, Hayato.

In: Advances in Mathematics, Vol. 273, 09.03.2015, p. 324-379.

Research output: Contribution to journalArticle

@article{0a4e986ee6c045b9b28ec71d74d94458,
title = "A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions",
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 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.",
author = "Hayato Chiba",
year = "2015",
month = "3",
day = "9",
doi = "10.1016/j.aim.2015.01.001",
language = "English",
volume = "273",
pages = "324--379",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

TY - JOUR

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

AU - Chiba, Hayato

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.

UR - http://www.scopus.com/inward/record.url?scp=84921327052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84921327052&partnerID=8YFLogxK

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 -