Mathematische Annalen

, Volume 344, Issue 3, pp 703–716

Loewner matrices and operator convexity


    • Indian Statistical Institute
  • Takashi Sano
    • Department of Mathematical Sciences, Faculty of ScienceYamagata University

DOI: 10.1007/s00208-008-0323-3

Cite this article as:
Bhatia, R. & Sano, T. Math. Ann. (2009) 344: 703. doi:10.1007/s00208-008-0323-3


Let f be a function from \({\mathbb{R}_{+}}\) into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form \({\left [\frac{f(p_i) - f(p_j)}{p_i-p_j}\right ]_{\vphantom {X_{X_1}}}}\) are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f (t) = tg(t) for some operator convex function g if and only if these matrices are conditionally positive definite. Elementary proofs are given for the most interesting special cases f (t) = tr , and f (t) = t log t. Several consequences are derived.

Mathematics Subject Classification (2000)


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© Springer-Verlag 2008