Publications and Preprints

Loewner Matrices and Operator Convexity
Rajendra Bhatia and Takashi Sano
Let $f$ be a function from $\mathbb{R}_{+}$ into itself. A classic theorem of K. L\"owner says that $f$ is operator monotone if and only if all matrices of the form $\left [\frac{f(p_i) - f (p_j)}{p_i-p_j} \right ]$ 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.

isid/ms/2008/07 [fulltext]

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