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

2013/04/30 by Pattrawut Chansangiam, Chansangiam, Pattrawut
Mathematics · #28A25 #47A63 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:28A25 #msc:47A63

paper · pdf · doi:10.48550/arxiv.1304.7936

9 pages

arxiv created 2013/04/30 · arxiv updated 2013/05/01

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 ↦ 1 !_t x for t ∈ [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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