Bhatia, Rajendra ; Sano, Takashi (2009) Loewner matrices and operator convexity Mathematische Annalen, 344 (3). pp. 703716. ISSN 00255831

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Official URL: http://www.springerlink.com/index/c66075737u068842...
Related URL: http://dx.doi.org/10.1007/s0020800803233
Abstract
Let f be a function from R+ into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form [f(p_{i})f(p_{j})/p_{i}p_{j}] 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.
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Deposited On:  08 Oct 2010 06:57 
Last Modified:  16 May 2016 13:32 
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