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arXiv · 1304.7936

Integral Representations and Decompositions of Operator Monotone Functions on the Nonnegative Reals

Pattrawut Chansangiam
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Abstract

In this paper, we show that there is a one-to-one correspondence between operator monotone functions on the nonnegative reals and finite Borel measures on the unit interval. This correspondence appears as an integral representation of special operator monotone functions $x \mapsto 1\,!_t\,x$ for $t \in [0,1]$ with respect to a finite Borel measure on $[0,1]$, here $!_t$ denotes the $t$-weighted harmonic mean. Hence such functions form building blocks for arbitrary operator monotone functions on the nonnegative reals. Moreover, we use this integral representation to decompose operator monotone functions.

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Pattrawut Chansangiam. 2013-04-30. Integral Representations and Decompositions of Operator Monotone Functions on the Nonnegative Reals. https://arxiv.org/abs/1304.7936

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