Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables
arXiv:1309.1791
Abstract
We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations which reflect their asymptotic behavior at infinity. Loewner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Loewner'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.
53 pages
References in corpus (6)
- Truncated K-moment problems in several variables
- Noncommutative ball maps
- Global holomorphic functions in several non-commuting variables II
- Operator monotone functions of several variables
- Nevanlinna representations in several variables
- Boundary behavior of analytic functions of two variables via generalized models