Properties of total sequences of operators and some of their applications
type: Item
instance of: scholarly article
title: Properties of total sequences of operators and some of their applications
zbMATH Open document ID: 0569.47039
DOI: 10.1007/BF01141774
author: E. N. Domanskij
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Let X and Y be normed linear spaces. A sequence of linear operators \(\{A_ n\}\), each \(A_ n\) acting on X into Y, is said to be total if \(\cap^{\infty}_{n=1}Ker A_ n=\{0\}.\) A sequence of bounded linear operators \(\{B_ n\}\) \((n=1,2,...)\) is said to be dually total if the sequence of dual operators \(\{B_ n^*\}\) is total. In this paper the author characterized total and dually total sequences in terms of the fundamentalness of certain linear sets and formulated some necessary conditions for the totalness, dual totalness of sequences of operators. Using these results the author obtained necessary conditions for the linear regularizability of the inverse operator \(A^{-1}\), where \(A\in B_ 0(X,Y)(B_ 0(X,Y)\) is the set of all bounded injective operators). And also the author gives two explicit examples of \(B_ 0(X,Y)\) whose inverse operators are not linearly regularizable, namely that \(B_ 0(C^*[0,1],L_{\infty}[0,1])\) and \(B_ 0(\ell_ 2(\Gamma),\ell_{\infty}(\Gamma)),\) \(\Gamma\) is nonseparable.
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 46A32
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 3909163
zbMATH Keywords: total and dually total sequences in terms of the fundamentalness of certain linear sets
zbMATH Keywords: linear regularizability of the inverse operator
reviewed by: Rho Jae-Chul
MaRDI profile type: Publication
cites work: On measures in independent fields (edited by S. Hartman)
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cites work: Classical Banach spaces
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full work available at URL: https://doi.org/10.1007/bf01141774
OpenAlex ID: W2012535264
description: scientific article; zbMATH DE number 3909163