On Schwarz type inequalities
Authors:
K. Tanahashi, A. Uchiyama and M. Uchiyama
Journal:
Proc. Amer. Math. Soc. 131 (2003), 25492552
MSC (2000):
Primary 47A30, 47A63, 47B15
Published electronically:
November 27, 2002
MathSciNet review:
1974654
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Additional Information
Abstract: We show Schwarz type inequalities and consider their converses. A continuous function is said to be semioperator monotone on if is operator monotone on . Let be a bounded linear operator on a complex Hilbert space and be the polar decomposition of . Let and for . (1) If a nonzero function is semioperator monotone on , then for , where . (2) If are semioperator monotone on , then for . Also, we show converses of these inequalities, which imply that semioperator monotonicity is necessary.
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M. Uchiyama, Further extension of the HeinzKatoFuruta inequality, Proc. Amer. Math. Soc., 127 (1999), 28992904. MR 2000a:47045
 1.
 J. S. Aujla, Perturbation bounds for certain operator functions, Math. Inequal. Appl., 4 (2001), 609617. MR 2002h:47015
 2.
 J. I. Fujii and M. Fujii, An analogue to Hansen's theory of generalized Löwner's functions, Math. Japonica, 35 (1990), 327330. MR 91d:47015
 3.
 T. Furuta, An extension of the HeinzKato theorem, Proc. Amer. Math. Soc., 120 (1994), 785787. MR 94e:47034
 4.
 E. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwer's theorem, Math. Ann., 258 (1982), 229241.
 5.
 E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415438. MR 13:471f
 6.
 T. Kato, Notes on some inequalities for linear operators, Math. Ann., 125 (1952), 208212. MR 14:766e
 7.
 M. Uchiyama, Further extension of the HeinzKatoFuruta inequality, Proc. Amer. Math. Soc., 127 (1999), 28992904. MR 2000a:47045
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Additional Information
K. Tanahashi
Affiliation:
Department of Mathematics, Tohoku Pharmaceutical University, Sendai 9818558, Japan
Email:
tanahasi@tohokupharm.ac.jp
A. Uchiyama
Affiliation:
Mathematical Institute, Tohoku University, Sendai 9808578, Japan
Email:
uchiyama@math.tohoku.ac.jp
M. Uchiyama
Affiliation:
Department of Mathematics, Fukuoka University of Education, Munakata 8114192, Japan
Email:
uchiyama@fukuokaedu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993902068892
PII:
S 00029939(02)068892
Keywords:
Schwarz inequality,
HeinzKatoFuruta inequality
Received by editor(s):
December 17, 2001
Received by editor(s) in revised form:
March 29, 2002
Published electronically:
November 27, 2002
Additional Notes:
This research was supported by GrantinAid Research No. 12640187.
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2002
American Mathematical Society
