A unified approach to operator monotone functions

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.