On a class of operator monotone functions of several variables

Abstract

The classical Lowner theorem [1]–[3] (see also [4, Remarks to Sec. X.2]) states that a continuous function f : (0,+∞) → R possesses the property f(A) ≤ f(B) for all pairs of bounded self-adjoint operators A and B in the Hilbert space H such that 0 < A ≤ B if and only if f can be analytically continued to the domain C \ (−∞, 0] and this continuation maps the open upper half-plane into its closure. Such functions are called operator monotone functions of one variable. They find important applications in calculus, mathematical physics, and the theory of electric networks (see [3], [5] and the bibliography there). Note that, in view of the maximum principle, an operator monotone function of one variable either maps the open upper half-plane into itself or is a real constant.