The spectral characterization of generalized projections
type: Item
instance of: scholarly article
title: The spectral characterization of generalized projections
zbMATH Open document ID: 1067.47001
author: Hongke Du
author: Yuan Li
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
review text: A bounded operator \(T\) on a Hilbert space \(H\) is called a generalized projection if \(T^2=T^*\). This notion was first introduced in the finite-dimensional case by \textit{J. Groß} and \textit{G. Trenkler} [Linear Algebra Appl. 264, 463--474 (1997; Zbl 0887.15024)]. The authors of the paper under review prove that an operator \(T\) is a generalized projection if and only if sp\((T)\subseteq\{0,1,e^{\pm i\frac{2}{3}\pi}\}\). They also extend the main result of \textit{J. K. Baksalary} and \textit{X. Liu} [Linear Algebra Appl. 388, 61--65 (2004; Zbl 1081.15018)] to the infinite-dimensional case. In fact, they introduce a spectral representation for a generalized projection and then show that the following statements are equivalent: (i) \(T\) is a generalized projection; (ii) \(T^4=T\) and \(T\) is normal; (iii) \(T^4=T\) and \(T\) is a partial isometry.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 2173032
zbMATH Keywords: generalized projection
zbMATH Keywords: spectral resolution
zbMATH Keywords: normal operator
zbMATH Keywords: partial isometry
zbMATH Keywords: spectral representation
MaRDI profile type: Publication
full work available at URL: j.laa.2004.11.027
OpenAlex ID: W2002073810
cites work: On linear combinations of generalized projectors
cites work: An alternative characterization of generalized projectors
cites work: Q3999065
cites work: Generalized and hypergeneralized projectors
cites work: Q3667432
DOI: 10.1016/J.LAA.2004.11.027
description: scientific article; zbMATH DE number 2173032