Abstract
It is shown that non-negative, increasing, convex (respectively, concave) functions are superadditive (respectively, subadditive) with respect to submajorisation on the positive cone of the space of all τ-measurable operators affiliated with a semifinite von Neumann algebra. This extends recent results for n × n-matrices by Ando-Zhan, Kosem and Bourin-Uchiyama.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Ando, Comparison of norms‖ |f(A) − f(B)| ‖ and ‖ | f(|A − B|) ‖ , Math. Z. 197 (1988), 403–409.
T. Ando, X. Zhan, Norm inequalities related to operator monotone functions, J. Math. Ann. 315 (1999), 771–780.
J.S. Aujla, F.C. Silva, Weak majorisation inequalities and convex functions, Linear Algebra Appli. 369 (2003), 217–233.
J.C. Bourin, M. Uchiyama, A matrix subadditivity inequality for f(A + B) and f(A) + f(B), Linear Algebra Appli. 423 (2007), 512–518.
V.I. Chilin, F.A. Sukochev, The triangle inequality for operators that are measurable with respect to Hardy–Littlewood order, Izv. Akad. nauk UzSSR Ser. Fiz.-Mat. Nauk. 104–105(4), (1988) 44–50 (Russian).
J. Dixmier, Von Neumann Algebras, Mathematical Library, vol 27. North Holland, Amsterdam (1981).
P.G. Dodds, T.K. Dodds, On a submajorization inequality of T. Ando, Operator Theory: Adv Appl 75 (1995), 113–131.
P.G. Dodds, T.K.-Y. Dodds, B. de Pagter, Noncommutative Banach function spaces, Math. Zeit., 201 (1989), 583–597.
P.G. Dodds, T.K.-Y. Dodds, B. de Pagter, Noncommutative Köthe duality, Trans. Amer. Math. Soc., 339 (1993), 717–750.
T. Fack, Sur la notion de valeur caractéristisque, J. Operator Theory, 7 (1982), 307–333.
T. Fack, H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific J. Math., 123 (1986), 269–300.
T. Kosem, Inequalities between ‖ f(A + B) ‖ and ‖f(A) + f(B) ‖, Linear Algebra Appl., 418 (2006), 153–160.
M. Uchiyama, Subadditivity of eigenvalue sums, Proc. Amer. Math. Soc., 134 (2005), 1405–1412.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the Australian Research Council.
Rights and permissions
About this article
Cite this article
Dodds, P.G., Sukochev, F.A. Submajorisation inequalities for convex and concave functions of sums of measurable operators. Positivity 13, 107–124 (2009). https://doi.org/10.1007/s11117-008-2206-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-008-2206-y