On Neumann operators
type: Item
instance of: scholarly article
title: On Neumann operators
zbMATH Open document ID: 0870.47001
author: T. Bermúdez
author: Antonio Martinón
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-09-15T00:00:00Z
review text: Let \(X\) denote a complex Banach space and \(L(X)\) the set of bounded linear operators defined on \(X\) with values in \(X\). An operator \(T\in L(X)\) is called a Neumann operator if for each \(x\in X\), the Neumann series \(\sum^\infty_{n=0} T^nx\) converges whenever \(\lim_{n\to\infty} T^nx=0\). It was shown by \textit{J. Schulz} [Beitr. Anal. 12, 177-183 (1978; Zbl 0425.47005)] that if some power of \(T\) is strictly singular or strictly cosingular, then \(T\) is a Neumann operator. More recently, \textit{D. Medkova} [Czech. Math. J. 41(116), No. 2, 312-316 (1991; Zbl 0754.47005)] proved that \(T\) is a Neumann operator if its distance from the set of all compact operators is less than 1. Other examples of such operators are projections and quasi-Riesz operators. In the paper under review the authors show that if there exists a nonnegative integer \(k\) such that the range \(R[(I-T)^k]\) is closed and the restriction of \(I-T\) to \(R[(I-T)^k]\) is injective, then \(T\) is a Neumann operator. This result is used to derive various sufficient conditions previously obtained by other authors, including the two referred to above.
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 908600
zbMATH Keywords: Neumann operator
zbMATH Keywords: Neumann series
zbMATH Keywords: strictly singular
zbMATH Keywords: strictly cosingular
zbMATH Keywords: quasi-Riesz operators
reviewed by: R. W. Cross
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0233
OpenAlex ID: W2010354537
DOI: 10.1006/JMAA.1996.0233
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description: scientific article; zbMATH DE number 908600