Loewner's theorem in several variables

In this paper we establish a multivariable non-commutative generalization of Loewner's theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half-plane into itself. The non-commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples, not a priori assumed to be analytic or continuous, as a Schur complement of a positive linear matrix pencil. To achieve this we establish the abstract integral formula for these functions using only matrix convexity and LMIs that represents operator monotone and operator concave free functions as a conditional expectation of a Schur complement of a linear matrix pencil on a tensor product operator algebra. The results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so, including the Karcher mean, to obtain an explicit, closed formula for these operator means.