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On some norm inequalities involving the commutator and XY - YXT
Che-Man Cheng1; Kin-Sio Fong2; Weng-Fai Lei1
2013-03-15
Source PublicationLinear Algebra and Its Applications
ISSN0024-3795
Volume438Issue:6Pages:2793-2807
Abstract

Let X and Y be any n × n complex matrices, ∥ ·∥ be the Frobenius norm and ∥·∥ be the (2,2)-norm defined by ∥X∥=s (X)+s(X), where s (X)≥⋯≥ sn are the singular values of X. Based on the paper [A. Böttcher and D. Wenzel, The Frobenius norm and the commutator, Linear Algebra Appl. 429 (2008) 1864-1885], a common approach is found for proving the following three inequalities: ∥XY-YX∥ ≤ √2∥X∥∥Y∥, ∥XY-YX∥≤ √2∥X∥ ∥Y∥and∥XY- YX∥≤ √2∥X∥∥Y∥ . The first two are known and the third one is new. Their equality cases are also determined. In particular it is found that, while there are common analogous equality cases, the third inequality admits an equality case that the other two do not have as a corresponding counterpart. 

KeywordNorm Inequality Commutator
DOIhttps://doi.org/10.1016/j.laa.2012.11.007
URLView the original
Indexed BySCIE
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000314321000014
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Cited Times [WOS]:3   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorChe-Man Cheng
Affiliation1.Department of Mathematics, University of Macau, Macao, China
2.Department of Finance and Business Economics, University of Macau, Macao, China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Che-Man Cheng,Kin-Sio Fong,Weng-Fai Lei. On some norm inequalities involving the commutator and XY - YXT[J]. Linear Algebra and Its Applications,2013,438(6):2793-2807.
APA Che-Man Cheng,Kin-Sio Fong,&Weng-Fai Lei.(2013).On some norm inequalities involving the commutator and XY - YXT.Linear Algebra and Its Applications,438(6),2793-2807.
MLA Che-Man Cheng,et al."On some norm inequalities involving the commutator and XY - YXT".Linear Algebra and Its Applications 438.6(2013):2793-2807.
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