Operator monotone functions and Löwner functions of several variables


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.


Jim Agler

Department of Mathematics, UC San Diego, 9500 Gilman Drive #0112, La Jolla, CA 92093-0112

John E. McCarthy

Mathematics Department, Washington University in St. Louis, St. Louis, MO 63130 and
School of Mathematics, Trinity College, Dublin 2, Ireland

N. J. Young

School o Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom, and
School of Mathematics & Statistics, Newcastle University, Newcastle upon Tyne NE3 4LR, United Kingdom