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TRACE INEQUALITIES AND A QUESTION OF BOURIN

Part of: General theory of linear operators Special classes of linear operators Basic linear algebra

Published online by Cambridge University Press:  25 January 2013

SAJA HAYAJNEH  and
FUAD KITTANEH
SAJA HAYAJNEH
Affiliation:
Department of Mathematics, Irbid National University, Irbid 2600, Jordan email [email protected]
FUAD KITTANEH*
Affiliation:
Department of Mathematics, University of Jordan, Amman, Jordan
*
✉[email protected]
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Abstract

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We give an affirmative answer to one of the questions posed by Bourin regarding a special type of inequality referred to as subadditivity inequalities in the case of the Hilbert–Schmidt and the trace norms. We formulate the solution for arbitrary commuting positive operators, and we conjecture that it is true for all unitarily invariant norms and all commuting positive operators. New related trace inequalities are also presented.

Keywords

Unitarily invariant normoperator inequalityconcave functiontrace normHilbert–Schmidt normpositive operator

MSC classification

Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A30: Norms (inequalities, more than one norm, etc.) 47A63: Operator inequalities 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 384 - 389
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Ando, T. and Zhan, X., ‘Norm inequalities related to operator monotone functions’, Math. Ann. 315 (1999), 771–780.Google Scholar
Bhatia, R., Matrix Analysis (Springer, New York, 1997).CrossRefGoogle Scholar
Bhatia, R. and Kittaneh, F., ‘Norm inequalities for positive operators’, Lett. Math. Phys. 43 (1998), 225–231.Google Scholar
Bourin, J. C., ‘Matrix subadditivity inequalities and block-matrices’, Internat. J. Math. 20 (2009), 679–691.Google Scholar
Bourin, J. C., ‘A matrix subadditivity inequality for symmetric norms’, Proc. Amer. Math. Soc. 138 (2010), 495–504.CrossRefGoogle Scholar
Bourin, J. C. and Uchiyama, M., ‘A matrix subadditivity inequality for $f(A+ B)$ and $f(A)+ f(B)$’, Linear Algebra Appl. 423 (2007), 512–518.CrossRefGoogle Scholar
Kittaneh, F., ‘A note on the arithmatic-geometric mean inequality for matrices’, Linear Algebra Appl. 171 (1992), 1–8.Google Scholar
Kosem, T., ‘Inequalities between $\Vert f(A+ B)\Vert $ and $\Vert f(A)+ f(B)\Vert $’, Linear Algebra Appl. 418 (2006), 153–160.Google Scholar