Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 273, Issue 1, Pages 283-328Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2017.04.001
Keywords
Free analysis; Noncommutative functions; Operator representations; Matrix monotonicity
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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.
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