Translation-invariant maps and applications
type: Item
instance of: scholarly article
title: Translation-invariant maps and applications
zbMATH Open document ID: 1167.15021
author: M. Niezgoda
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-13T00:00:00Z
review text: A general framework for deriving upper bounds for some quantities induced by translation-invariant maps is given. Vectorial intervals induced by two preorders on a linear space and radii of families of maps at a point are used. Section 2 introduces the relevant notation and terminology and it includes the main general result, Theorem 2.2. At the end of Section 2 the study is confined to GIC preorders; an extension of a result given by \textit{C.-K. Li} and \textit{R. Mathias} [J. Inequal. Appl. 3, 137--142 (1999; Zbl 0958.15017)] is proved. Section 3 is devoted to bounds for norms of the difference of two operators. In particular, norms of commutators are considered. For instance, recent results given by \textit{Y.-Q. Wang} and \textit{H.-K. Du} [J. Math. Anal. Appl. 342, 747--751 (2008; Zbl 1139.47010)] and \textit{R. Bhatia} and \textit{F. Kittaneh} [Oper. Matrices 2, 143--151 (2008; Zbl 1147.15019)] are corollaries of general results in this section. The last section deals with the Grüss inequality. Again, a general theorem is proved which has some known results as corollaries. The paper is well written. It seems that the study of general results and their applications may give some results that are not included in the paper.
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 5553657
zbMATH Keywords: translation-invariant map
zbMATH Keywords: preorder
zbMATH Keywords: vectorial interval
zbMATH Keywords: numerical radius
zbMATH Keywords: G(c)-range
zbMATH Keywords: G-ma\-jo\-ri\-za\-tion
zbMATH Keywords: commutator
zbMATH Keywords: G-contractive map
zbMATH Keywords: Grüss type inequality
zbMATH Keywords: norm bounds
zbMATH Keywords: eigenvalue
zbMATH Keywords: singular value
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.12.041
OpenAlex ID: W1989143351
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DOI: 10.1016/J.JMAA.2008.12.041
description: scientific article; zbMATH DE number 5553657