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Jensen's operator inequality for functions of several variables

Author(s): Huzihiro Araki; Frank Hansen
Journal: Proc. Amer. Math. Soc. 128 (2000), 2075-2084.
MSC (1991): Primary 47A63; Secondary 47A80, 47Bxx
Posted: February 16, 2000
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Abstract: The operator convex functions of several variables are characterized in terms of a non-commutative generalization of Jensen's inequality, extending previous results of F. Hansen and G.K. Pedersen for functions of one variable and of F. Hansen for functions of two variables.

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T. Ando. Concavity of certain maps of positive definite matrices and applications to adamard products. Linear Algebra Appl., 26:203-241, 1979. MR 80f:15023

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MR 94h:15008 J.S. Aujla. Matrix convexity of functions of two variables. Linear Algebra and Its Applications, 194:149-160, 1993. MR 94h:15008

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P.R. Halmos. A Hilbert space problem book. Van Nostrand, 1967. MR 34:8178
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F. Hansen. An operator inequality. Math. Ann., 246:249-250, 1980. MR 82a:46065

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F. Hansen. Jensen's operator inequality for functions of two variables. Proc. Amer. Math. Soc., 125:2093-2102, 1997. MR 97i:47027

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F. Hansen. Operator convex functions of several variables. Publ. Res. Inst. Math. Sci., 33:443-463, 1997. MR 99b:47022b; MR 99b:47022a

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F. Hansen and G.K. Pedersen. Jensen's inequality for operators and öwner's theorem. Math. Ann., 258:229-241, 1982. MR 83g:47020

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A. Korányi. On some classes of analytic functions of several variables. Trans Amer. Math. Soc., 101:520-554, 1961. MR 25:226

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G.K. Pedersen. Extreme $ n $-tuples of elements in $C^* $-algebras. Bull. London Math. Soc., 19:264-270, 1987. MR 88m:46070

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Additional Information:

Huzihiro Araki
Affiliation: Department of Mathematics, Faculty of Science and Technology, The Science University of Tokyo, Noda, Chiba-ken 278-8510, Japan
Email: araki@ma.noda.sut.ac.jp

Frank Hansen
Affiliation: Institute of Economics, University of Copenhagen, Studiestraede 6, 1455 Copenhagen K, Denmark
Email: frank.hansen@econ.ku.dk

DOI: 10.1090/S0002-9939-00-05371-5
PII: S 0002-9939(00)05371-5
Received by editor(s): August 30, 1998
Posted: February 16, 2000
Communicated by: David R. Larson
Copyright of article: Copyright 2000, American Mathematical Society

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