A unified approach to operator monotone functions

Abstract

The notion of operator monotonicity dates back to a work by Löwner in 1934. A map F:Sn→Sm is called operator monotone, if A⪰B implies F(A)⪰F(B). (Here, Sn is the space of symmetric matrices with the semidefinite partial order ⪰.) Often, the function F is defined in terms of an underlying simpler function f. Of main interest is to find the properties of f that characterize operator monotonicity of F. In that case, it is said that f is also operator monotone. Classical examples are the Löwner operators and the spectral (scalar-valued isotropic) functions. Operator monotonicity for these two classes of functions is characterized in seemingly very different ways.This work extends the notion of operator monotonicity to symmetric functions f on k arguments. The latter is used to define (generated) k-isotropic mapsF:Sn→S(nk) for any n≥k. Necessary and sufficient conditions are given for f to generate an operator monotone k-isotropic map F. When k=1, the k-isotropic map becomes a Löwner operator and when k=n it becomes a spectral function. This allows us to reconcile and explain the differences between the conditions for monotonicity for the Löwner operators and the spectral functions.