On Schwarz type inequalities

Authors:
K. Tanahashi, A. Uchiyama and M. Uchiyama

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2549-2552

MSC (2000):
Primary 47A30, 47A63, 47B15

DOI:
https://doi.org/10.1090/S0002-9939-02-06889-2

Published electronically:
November 27, 2002

MathSciNet review:
1974654

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Abstract: We show Schwarz type inequalities and consider their converses. A continuous function is said to be semi-operator 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 non-zero function is semi-operator monotone on , then for , where . (2) If are semi-operator monotone on , then for . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.

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Additional Information

**K. Tanahashi**

Affiliation:
Department of Mathematics, Tohoku Pharmaceutical University, Sendai 981-8558, Japan

Email:
tanahasi@tohoku-pharm.ac.jp

**A. Uchiyama**

Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Email:
uchiyama@math.tohoku.ac.jp

**M. Uchiyama**

Affiliation:
Department of Mathematics, Fukuoka University of Education, Munakata 811-4192, Japan

Email:
uchiyama@fukuoka-edu.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-02-06889-2

Keywords:
Schwarz inequality,
Heinz-Kato-Furuta 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 Grant-in-Aid Research No. 12640187.

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2002
American Mathematical Society