Loewner matrices and operator convexity

Bhatia, Rajendra ; Sano, Takashi (2009) Loewner matrices and operator convexity Mathematische Annalen, 344 (3). pp. 703-716. ISSN 0025-5831

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Official URL: http://www.springerlink.com/index/c66075737u068842...

Related URL: http://dx.doi.org/10.1007/s00208-008-0323-3


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(pi)-f(pj)/pi-pj] 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.

Item Type:Article
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ID Code:2552
Deposited On:08 Oct 2010 06:57
Last Modified:16 May 2016 13:32

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