Mathematische Annalen

, Volume 344, Issue 3, pp 703–716 | Cite as

Loewner matrices and operator convexity

  • Rajendra BhatiaEmail author
  • Takashi Sano


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) = t g(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) = t r , and f (t) = t log t. Several consequences are derived.

Mathematics Subject Classification (2000)

15A48 47A63 42A82 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Indian Statistical InstituteNew DelhiIndia
  2. 2.Department of Mathematical Sciences, Faculty of ScienceYamagata UniversityYamagataJapan

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