Spectral properties of the operator associated with a retarded functional differential equation in Hilbert space
type: Item
instance of: scholarly article
title: Spectral properties of the operator associated with a retarded functional differential equation in Hilbert space
zbMATH Open document ID: 0699.34071
author: Jin-Mun Jeong
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: Under the assumptions that \((V,\|.\|)\) is a Hilbert space densely and continuously imbedded in a Hilbert space H, the imbedding \(V\subset H\) is compact, \(-A_ 0\) is the operator associated with a bounded sesquilinear form a(u,v) defined in \(V\times V\) and satisfying Gårding's inequality Re a(u,v)\(\geq c\| u\|^ 2\), \(\gamma\in R\) and the real function a(s) is Hölder continuous, the spectral properties of the infinitesimal generator A of the solution semigroup S(t) for \[ \frac{du}{dt}=A_ 0u(t)+\gamma A_ 0u(t-h)+\int^{0}_{- h}a(s)A_ 0u(t+s)ds \] are investigated. If \(m(\lambda)=1+\gamma e^{- \lambda h}+\int^{0}_{-h}e^{\lambda s}a(s)ds\) and \(m(0)=0\), then 0 is an eigenvalue of A with infinite multiplicity. 0 is an isolated point of \(\sigma\) (A) if it is a simple zero of m(\(\lambda)\), and it is a cluster point of \(\sigma\) (A) if it is a multiple zero of m(\(\lambda)\). If m(0)\(\neq 0\), \(\gamma\neq 0\) and the generalized eigenvectors of \(A_ 0\) are complete in H, then the generalized eigenvectors of A are complete in \(M=H\times L^ 2(-h,0;V)\).
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 4146483
zbMATH Keywords: identification problems
zbMATH Keywords: Hilbert space
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: solution semigroup
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenvectors
MaRDI profile type: Publication
full work available at URL: pjaa.65.98
OpenAlex ID: W1982343507
Wikidata QID: Q115219937
cites work: Completeness and F-completeness of eigenfunctions associated with retarded functional differential equations
cites work: Structural properties of functional differential equations in Banach spaces
cites work: Q3810201
cites work: Structural operators for linear delay-differential equations in Hilbert space
DOI: 10.3792/PJAA.65.98
description: scientific article; zbMATH DE number 4146483