Strengthened convexity of positive operator monotone decreasing functions

  • Megumi Kirihata
  • Makoto Yamashita

Abstract

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function.

References

Ando, T., Topics on operator inequalities, Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University, Sapporo, 1978.

Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203–241. https://doi.org/10.1016/0024-3795(79)90179-4

Ando, T. and Hiai, F., Operator log-convex functions and operator means, Math. Ann. 350 (2011), no. 3, 611–630. https://doi.org/10.1007/s00208-010-0577-4

Aujla, J. S., Singh Rawla, M., and Vasudeva, H. L., Log-convex matrix functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 19–32.

Bhatia, R., Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4612-0653-8

Bhatia, R., Positive definite matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007.

Carlen, E. A., Frank, R. L., and Lieb, E. H., Some operator and trace function convexity theorems, Linear Algebra Appl. 490 (2016), 174–185. https://doi.org/10.1016/j.laa.2015.11.006

Choi, M. D., A Schwarz inequality for positive linear maps on $C^\ast $-algebras, Illinois J. Math. 18 (1974), 565–574.

Hansen, F. and Pedersen, G. K., Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1981/82), no. 3, 229–241. https://doi.org/10.1007/BF01450679

Hiai, F., Concavity of certain matrix trace and norm functions, Linear Algebra Appl. 439 (2013), no. 5, 1568–1589. https://doi.org/10.1016/j.laa.2013.04.020

Hiai, F., Concavity of certain matrix trace and norm functions. II, Linear Algebra Appl. 496 (2016), 193–220. https://doi.org/10.1016/j.laa.2015.12.032

Kian, M. and Dragomir, S. S., $f$-divergence functional of operator log-convex functions, Linear Multilinear Algebra 64 (2016), no. 2, 123–135. https://doi.org/10.1080/03081087.2015.1025686

Kubo, F. and Ando, T., Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224. https://doi.org/10.1007/BF01371042

Lieb, E. H., Convex trace functions and the Wigner-Yanase-Dyson conjecture, Advances in Math. 11 (1973), 267–288. https://doi.org/10.1016/0001-8708(73)90011-X
Published
2020-09-03
How to Cite
Kirihata, M., & Yamashita, M. (2020). Strengthened convexity of positive operator monotone decreasing functions. MATHEMATICA SCANDINAVICA, 126(3), 559-567. https://doi.org/10.7146/math.scand.a-120579
Section
Articles