Free Pick functions: Representations, asymptotic behavior and matrix monotonicity in several noncommuting variables

R. Nevanlinna showed that elements of the Pick class, the set of complex analytic functions taking the upper half plane into itself, have certain integral representations which reflect their asymptotic behavior at infinity. Later, Lowner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Lowner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus. (C) 2017 Elsevier Inc. All rights reserved.