On the spectral properties of differential operators with unbounded operator coefficients determined by a linear relation
type: Item
instance of: scholarly article
title: On the spectral properties of differential operators with unbounded operator coefficients determined by a linear relation
zbMATH Open document ID: 1239.47039
DOI: 10.1134/S0001434611010287
author: V. B. Didenko
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Let \(X\) be a Banach space over \(\mathbb{R}\) or \(\mathbb{C}\) and let \({\mathcal{U}}: [a,b]\times[a,b]\to \text{End}(X)\) a map defining a family of (strongly continuous) evolution operators on \([a,b]\). If \(\Gamma\) is a linear relation on \(X,\) the equation \(x(t)={\mathcal{U}}(t,a)x(a)+\int_a^t{\mathcal{U}}(t,s)f(s)\,ds\) defines an operator \({\mathcal{L}}_{\Gamma}x=f\) for all continuous functions \(x:[a,b]\to X\) such that \((x(a),x(b))\in \Gamma\). In this interesting work, the author shows that, if the operator \({\mathcal{L}}_{\Gamma}\) has one of ten given spectral properties, then the operator \(\Gamma-{\mathcal{U}}(b,a)\) has the same spectral properties. In particular, he shows that, if one of them has the Fredholm property, then the other has the same property.
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47A06
Mathematics Subject Classification ID: 47A11
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 35F99
zbMATH DE Number: 5980767
zbMATH Keywords: linear relation
zbMATH Keywords: invertibility
zbMATH Keywords: Fredholm property
zbMATH Keywords: Banach space
zbMATH Keywords: Banach algebra
zbMATH Keywords: evolution operator
zbMATH Keywords: Fredholm relation
zbMATH Keywords: resolvent set
reviewed by: George Karakostas
MaRDI profile type: Publication
cites work: Discrete and continuous boundary problems
cites work: Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations
cites work: Spectral analysis of linear relations and degenerate operator semigroups
cites work: Q4345292
Recommended article: Q5035984
Recommended article: Spectral analysis of differential operators with unbounded operator coefficients on the half-line
Recommended article: Q5012932
Recommended article: Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations
Recommended article: Linear differential operators with unbounded operator coefficients and semigroups of bounded operators
Recommended article: Spectral analysis of differential operators with unbounded operator coefficients in weighted spaces of functions
Recommended article: Q4496356
Recommended article: Q3986279
Recommended article: Q4871448
Recommended article: Spectral analysis of higher order differential operators with unbounded coefficients
description: scientific article; zbMATH DE number 5980767