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

الأصل · EN

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!ₜx 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.