Mathematische Annalen

, Volume 344, Issue 3, pp 703–716

# Loewner matrices and operator convexity

Article

## Abstract

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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