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arXiv · 1009.3921

Operator monotone functions and Löwner functions of several variables

Jim Agler·John E. McCarthy·Nicholas J. Young
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Abstract

We prove generalizations of Löwner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.

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Jim Agler, John E. McCarthy, Nicholas J. Young. 2013-12-18. Operator monotone functions and Löwner functions of several variables. https://arxiv.org/abs/1009.3921

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