## A characterisation of anti-Löwner functions

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- by Koenraad M. R. Audenaert
- Proc. Amer. Math. Soc.
**139**(2011), 4217-4223 - DOI: https://doi.org/10.1090/S0002-9939-2011-10935-3
- Published electronically: April 26, 2011
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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.## References

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## Bibliographic 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
- Received by editor(s): October 20, 2010
- Published electronically: April 26, 2011
- Communicated by: Marius Junge
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 4217-4223 - MSC (2010): Primary 15A60
- DOI: https://doi.org/10.1090/S0002-9939-2011-10935-3
- MathSciNet review: 2823067