Abstract
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.
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FJOURNAL = {Linear Algebra and its Applications},
VOLUME = {328},
YEAR = {2001},
NUMBER = {1-3},
PAGES = {131--152},
ISSN = {0024-3795},
CODEN = {LAAPAW},
MRCLASS = {47A56 (15A45 26B99 47A60 47A63)},
MRNUMBER = {1823518},
MRREVIEWER = {Frank Hansen},
DOI = {10.1016/S0024-3795(00)00308-6},
ZBLNUMBER = {0990.15010},
} -
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@book{sto32,
author={Stone, Marshall Harvey},
TITLE={Linear Transformations in {H}ilbert Space and their Applications to Analysis},
YEAR = {1932},
SERIES = {Amer. Math. Soc. Colloq. Publ.},
VOLUME = {15},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {New York},
ZBLNUMBER = {0005.40003},
MRNUMBER = {1451877},
} -
[var74] T. N. Varopoulos, "On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory," J. Funct. Anal., vol. 16, pp. 83-100, 1974.
@article {var74, MRKEY = {0355642},
AUTHOR = {Varopoulos, N. Th.},
TITLE = {On an inequality of von {N}eumann and an application of the metric theory of tensor products to operators theory},
JOURNAL = {J. Funct. Anal.},
VOLUME = {16},
YEAR = {1974},
PAGES = {83--100},
MRCLASS = {47A25},
MRNUMBER = {0355642},
MRREVIEWER = {T. Ando},
ZBLNUMBER = {0288.47006},
}
Full Article