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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A characterisation of anti-Löwner functions


Author: Koenraad M. R. Audenaert
Journal: Proc. Amer. Math. Soc. 139 (2011), 4217-4223
MSC (2010): Primary 15A60
Published electronically: April 26, 2011
MathSciNet review: 2823067
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Abstract: According to a celebrated result by Löwner, a real-valued function $ f$ is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences $ L_f=\left(\frac{f(x_i)-f(x_j)}{x_i-x_j}\right)_{i,j=1}^N$, is positive semidefinite for every integer $ N>0$ and any choice of $ x_1,x_2,\ldots,x_N$. In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions $ g$ defined on $ (0,+\infty)$ for which the matrix of divided sums $ K_g=\left(\frac{g(x_i)+g(x_j)}{x_i+x_j}\right)_{i,j=1}^N$, which we call its anti-Löwner matrix, is positive semidefinite for every integer $ N>0$ and any choice of $ x_1,x_2,\ldots,x_N\in(0,+\infty)$. Such functions, which we call anti-Löwner functions, have applications in the theory of Lyapunov-type equations.


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Additional Information

Koenraad M. R. Audenaert
Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, United Kingdom
Email: koenraad.audenaert@rhul.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10935-3
PII: S 0002-9939(2011)10935-3
Keywords: Matrix monotone functions, Löwner matrices, Lyapunov equation
Received by editor(s): October 20, 2010
Published electronically: April 26, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.