Compact linear operators on functional spaces with two norms
type: Item
instance of: scholarly article
title: Compact linear operators on functional spaces with two norms
zbMATH Open document ID: 0914.47002
DOI: 10.1007/BF01238216
published in: Integral Equations and Operator Theory
publication date: 1999-06-23T00:00:00Z
review text: The original version of this paper (in Ukrainian) was published in Sbirnik Praz' Instituta Matem. Akademii Nauk, Ukrainsk. SSR, No. 9, 104-129 (1947). Some statement from this paper were later rediscovered by P. D. Lax and by J. A. Dieudonne. In Section 1 some general facts of linear operator theory in a space with two norms are established. Important results, as for instance the well-known Hilbert-Schmidt theorem on expansions with respect to eigenfunctions, Mercer's theorem etc. are obtained in this general context. In Section 2, the author develops the theory of symmetrizable operators. Section 3 contains a theorem which can be used to investigate the asymptotics of eigenvalues of a compact operator, with applications to the theory of integral equations with a symmetrizable kernel. Although written more than fifty years ago, the paper has not only the value of an historical document but it remains as modern and fresh as a masterpiece can be. An excellent idea to offer this translation to a large audience.
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 01A75
zbMATH DE Number: 1129329
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: space with two norms
zbMATH Keywords: Hilbert-Schmidt theorem
zbMATH Keywords: Mercer's theorem
zbMATH Keywords: symmetrizable operators
zbMATH Keywords: asymptotics of eigenvalues
zbMATH Keywords: compact operator
zbMATH Keywords: integral equations with a symmetrizable kernel
author: Mark Grigorevich Kreĭn
reviewed by: Florian-Horia Vasilescu
MaRDI profile type: Publication
cites work: Q5765878
full work available at URL: https://doi.org/10.1007/bf01238216
OpenAlex ID: W1987451836
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description: scientific article; zbMATH DE number 1129329