Separated spectra and operator inequalities

Abstract

The main goal of this article is to show that many inequalities that are not valid in operator theory become true if we add a separation condition on the spectra. The applications include showing how monotone functions behave like operator monotone functions and how the Choi-Davis inequality becomes valid for convex functions under this separation condition.

Keywords

• Operator order
• separated spectra
• convex functions
• operator monotone functions

References

1. [1] T. Ando and F. Hiai, Log-majorization and complementary Golden–Thompson type inequalities, Linear Alg. Appl. 197/198, 113–131, 1994.
2. [2] T. Ando and F. Hiai, Operator log-convex functions and operator means, Math. Ann. 350 (3), 611–630, 2011.
3. [3] T. Ando and X. Zhan, Norm inequalities related to operator monotone functions, Math. Ann. 315, 771–780, 1999.
4. [4] R. Bhatia, Matrix analysis, Springer-Verlag, New York, 1997.
5. [5] J.C. Bourin and M. Uchiyama, A matrix subadditivity inequality for $f (A+B)$ and $f (A)+ f (B)$, Linear Algebra Appl. 423, 512–518, 2007.
6. [6] T. Furuta, J. Micic Hot, J. Pecaric and Y. Seo, Mond-Pecaric Method in Operator Inequalities, Monographs in Inequalities 1, Element, Zagreb, 2005.
7. [7] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246, 205-224, 1980.
8. [8] M. Lin, On an operator Kantorovich inequality for positive linear maps, J. Math. Anal. Appl. 402, 127–132, 2013.

9. [9] J. Mićić, Z. Pavić and J. Pečarić, Jensen’s inequality for operators without operator convexity, Linear Algebra Appl. 434, 1228–1237, 2011.
10. [10] H.R. Moradi, Z. Heydarbeygi and M. Sababheh, Subadditive inequalities for operators, Math. Ineq. Appl. 23 (1), 317–327, 2020.
11. [11] R. Nakamoto and M. Nakamura, Operator mean and Kantorovich inequality, Math. Japon. 44 (3), 495–498, 1996.
12. [12] M. Uchiyama, Operator monotone functions, positive definite kernels and majorization, Proc. Amer. Math. Soc. 138 (11), 3985–3996, 2010.