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Subadditivity Inequalities for Compact Operators

Published online by Cambridge University Press:  20 November 2018

Jean-Christophe Bourin ,
Tetsuo Harada  and
Eun-Young Lee
Jean-Christophe Bourin
Affiliation:
Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besaçon, France e-mail: jcbourin@univ-fcomte.fr
Tetsuo Harada
Affiliation:
6-12-28-102 Tamura, Sawaraku, Fukuoka 814-0175, Japan e-mail: tharada1005@gmail.com
Eun-Young Lee
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea e-mail: eylee89@knu.ac.kr
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Abstract

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Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon $ term. It does not seem possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon $ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also emphasizes matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

Keywords

47A6315A45concave or convex functionHilbert spaceunitary orbitscompact operatorscompressionsmatrix inequalities
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 25 - 36
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Aujla, J. S. and Bourin, J.-C., Eigenvalue inequalities for convex and log-convex functions. Linear Algebra Appl. 424 (2007), 25–35. http://dx.doi.org/10.1016/j.laa.2006.02.027 Google Scholar
[2] Bourin, J.-C., Harada, T. and Lee, E.-Y., Strict type inequalities for Hilbert space operators. Unpublished manuscript.Google Scholar
[3] Bourin, J.-C. and Hiai, F., Norm and anti-norm inequalities for positive semi-definite matrices. Internat. J. Math. 63 (2011), 1121–1138. http://dx.doi.org/10.1142/S0129167X1100715X Google Scholar
[4] Bourin, J.-C., Jensen and Minkowski inequalities for operator means and anti-norms. Preprint, arxiv:1106.2213v3. Google Scholar
[5] Bourin, J.-C. and Lee, E.-Y., Concave functions of positive operators, sums, and congruences. J. Operator Theory 63 (2010), 151–157.Google Scholar
[6] Bourin, J.-C. and Lee, E.-Y., Unitary orbits of Hermitian operators with convex or concave functions. Preprint, arxiv:1109.2384v1. Google Scholar
[7] Harada, T. and Kosaki, H., On equality condition for trace Jensen inequality in semi-finite vonNeumann algebras. Internat. J. Math. 19 (2008), 481–501. http://dx.doi.org/10.1142/S0129167X08004753 Google Scholar
[8] Rotfel'd, S. Ju., The singular values of a sum of completely continuous operators. In: Topics in Mathematical Physics 3 (1969), 73–78.Google Scholar
[9] Simon, B., Trace Ideals and Their Applications. Second edition, Amer. Math. Soc., Providence, RI, 2005.Google Scholar