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Operator monotone functions and Löwner functions of several variables

Lookup NU author(s): Professor Jim Agler, Professor Nicholas Young, John Cunningham



We prove generalizations of Löwner’s results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna’s theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.

Publication metadata

Author(s): Agler J, McCathy JE, Young NJ

Publication type: Article

Publication status: Published

Journal: Annals of Mathematics

Year: 2012

Volume: 176

Issue: 3

Pages: 1783-1826

Print publication date: 01/11/2012

Date deposited: 05/05/2012

ISSN (print): 0003-486X

ISSN (electronic): 1939-8980

Publisher: Mathematical Sciences Publishers


DOI: 10.4007/annals.2012.176.3.7


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Funder referenceFunder name
DMS-0966845National Science Foundation
DMS-1068830National Science Foundation
EP/J004545/1Engineering and Physical Sciences Research Council