Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T06:35:54.344Z Has data issue: false hasContentIssue false
  • English
  • Français

OPERATOR QUASILINEARITY OF SOME FUNCTIONALS ASSOCIATED WITH DAVIS–CHOI–JENSEN’S INEQUALITY FOR POSITIVE MAPS

Part of: Basic linear algebra General theory of linear operators

Published online by Cambridge University Press:  19 October 2016

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
Mathematics, College of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we establish operator quasilinearity properties of some functionals associated with Davis–Choi–Jensen’s inequality for positive maps and operator convex or concave functions. Applications for the power function and the logarithm are provided.

Keywords

Jensen’s inequalityoperator convex functionsarithmetic mean–geometric mean inequalitysuperadditivitysubadditivity

MSC classification

Primary: 47A63: Operator inequalities
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 322 - 332
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Choi, M. D., ‘A Schwarz inequality for positive linear maps on C ∗ -algebras’, Illinois J. Math. 18 (1974), 565–574.Google Scholar
Dragomir, S. S., ‘Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces’, J. Inequal. Appl. 2010 (2010), article 496821, 15 pages.CrossRefGoogle Scholar
Dragomir, S. S., Operator Inequalities of the Jensen, Čebyšev and Grüss Type, Springer Briefs in Mathematics (Springer, New York, 2012).Google Scholar
Dragomir, S. S., ‘Some inequalities for relative operator entropy’, RGMIA Res. Rep. Coll. 18 (2015), article 145, 12 pages [online http://rgmia.org/papers/v18/v18a145.pdf].Google Scholar
Dragomir, S. S., ‘Further inequalities for relative operator entropy’, RGMIA Res. Rep. Coll. 18 (2015), article 160, 11 pages [online http://rgmia.org/papers/v18/v18a160.pdf].Google Scholar
Dragomir, S. S., Fujii, J. I. and Seo, Y., ‘Bounds for an operator concave function’, Electron. J. Linear Algebra 26 (2013), 192–200.CrossRefGoogle Scholar
Fujii, J. I. and Kamei, E., ‘Relative operator entropy in noncommutative information theory’, Math. Japon. 34(3) (1989), 341–348.Google Scholar
Fujii, J. I. and Kamei, E., ‘Uhlmann’s interpolational method for operator means’, Math. Japon. 34(4) (1989), 541–547.Google Scholar
Kim, I. H., ‘Operator extension of strong subadditivity of entropy’, J. Math. Phys. 53 (2012), article 122204, 3 pages [online http://dx.doi.org/10.1063/1.4769176].CrossRefGoogle Scholar
Kluza, P. and Niezgoda, M., ‘Inequalities for relative operator entropies’, Electron. J. Linear Algebra 27 (2014), 851–864.Google Scholar
Kubo, F. and Ando, T., ‘Means of positive operators’, Math. Ann. 264 (1980), 205–224.Google Scholar
Moslehian, M. S., Mirzapour, F. and Morassaei, A., ‘Operator entropy inequalities’, Colloq. Math. 130 (2013), 159–168.CrossRefGoogle Scholar
Nakamura, M. and Umegaki, H., ‘A note on the entropy for operator algebras’, Proc. Japan Acad. 37 (1961), 149–154.Google Scholar
Nikoufar, I., ‘On operator inequalities of some relative operator entropies’, Adv. Math. 259 (2014), 376–383.Google Scholar
Pečarić, J., Furuta, T., Mićić Hot, J. and Seo, Y., Mond–Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space (Element, Zagreb, 2005).Google Scholar