Denseness of the generalized eigenvectors of an H-S discrete operator
type: Item
instance of: scholarly article
title: Denseness of the generalized eigenvectors of an H-S discrete operator
zbMATH Open document ID: 0679.47014
DOI: 10.1016/0022-1236(89)90074-8
author: John Locker
author: Patrick M. Lang
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let T be a closed densely defined linear operator in a Hilbert space H. Assume that there exists a point \(\xi\in \rho (T)\) such that the resolvent \(R_{\xi}(T)\) is a Hilbert-Schmidt oprator. Then, T is called an H-S discrete operator. It is a result of Dunford and Schwartz that if \(R_{\lambda}(T)\) satisfies \(\| R_{\lambda}(T)\| =O(| \lambda |^{-N})\) as \(\lambda\) \(\to \infty\) along certain rays \(\gamma_ 1,...,\gamma_ 5\), then H is the closure of the subspace of H spanned by the generalized eigenvectors of T. In particular, this result applies if T is the differential operator in \(L^ 2[0,1]\) determined by \(\tau =-[d/dt]^ 2\) and regular boundary conditions. In the present paper the authors generalize the Dunford and Schwartz result so that irregular boundary conditions are covered. Their main theorem is the following: Let T be an H-S discrete operator. Let \(\sigma (T)=\{\lambda_ i\}^{\infty}_{i=1}\). Assume that there exists a set of five rays such that (i) the angles between adjacent rays are less than \(\pi\) /2; (ii) for \(| \lambda |\) sufficiently large all the points on the five rays belong to \(\rho\) (T), and \(\| R_{\lambda}(T)\|\) is bounded for these \(\lambda\) ; (iii) on at least one of the rays \(\| R_{\lambda}(T)\| \to 0\) as \(\lambda\) \(\to \infty.\) Then \(\overline{S_{\infty}}=\overline{S^*_{\infty}}=H\) and \(M_{\infty}=M^*_{\infty}=\{0\}.\) Here \(S_{\infty}\) and \(M_{\infty}\) are the subspaces of H consisting of all \(x\in H\) such that \(x=\sum^{\infty}_{i=1}P_ ix\) and such that \(P_ ix=0\) for \(i=1,2,...\), respectively. \(P_ i\) is the projection of H onto the generalized eigenspace \({\mathcal N}([\lambda_ iI- T]^{m_ i})\) along \({\mathcal R}([\lambda_ iI-T]^{m_ i}).\) \(m_ i\) is the ascent of \(\lambda_ iI-T\). \(S^*_{\infty}\) and \(M^*_{\infty}\) are defined similarly using \(P^*_ i\).
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 34L99
zbMATH DE Number: 4113336
zbMATH Keywords: Hilbert-Schmidt discrete operator
zbMATH Keywords: closed densely defined linear operator in a Hilbert space
zbMATH Keywords: resolvent
zbMATH Keywords: Hilbert-Schmidt oprator
zbMATH Keywords: H-S discrete operator
zbMATH Keywords: differential operator
zbMATH Keywords: irregular boundary conditions
zbMATH Keywords: generalized eigenspace
zbMATH Keywords: ascent
reviewed by: Richard C. Gilbert
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90074-8
OpenAlex ID: W1971027498
cites work: Q5611178
cites work: Q5523220
cites work: Q3285751
cites work: Spectral theory for Fredholm operators
cites work: Spectral theory of two-point differential operators determined by \(-D^ 2\). I: Spectral properties
cites work: Q5829957
cites work: Q3966611
description: scientific article; zbMATH DE number 4113336