Theoretical Statistics and Mathematics Unit, ISI Delhi

Loewner Matrices and Operator Convexity

by 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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