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

Published electronically:
November 27, 2002

MathSciNet review:
1974654

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Jaspal Singh Aujla,*Perturbation bounds for certain operator functions*, Math. Inequal. Appl.**4**(2001), no. 4, 609–617. MR**1859665**, 10.7153/mia-04-53**2.**Junichi Fujii and Masatoshi Fujii,*An analogue to Hansen’s theory of generalized Löwner’s functions*, Math. Japon.**35**(1990), no. 2, 327–330. MR**1049097****3.**Takayuki Furuta,*An extension of the Heinz-Kato theorem*, Proc. Amer. Math. Soc.**120**(1994), no. 3, 785–787. MR**1169027**, 10.1090/S0002-9939-1994-1169027-6**4.**E. Hansen and G. K. Pedersen,*Jensen's inequality for operators and Löwer's theorem,*Math. Ann.,**258**(1982), 229-241.**5.**Erhard Heinz,*Beiträge zur Störungstheorie der Spektralzerlegung*, Math. Ann.**123**(1951), 415–438 (German). MR**0044747****6.**Tosio Kato,*Notes on some inequalities for linear operators*, Math. Ann.**125**(1952), 208–212. MR**0053390****7.**Mitsuru Uchiyama,*Further extension of Heinz-Kato-Furuta inequality*, Proc. Amer. Math. Soc.**127**(1999), no. 10, 2899–2904. MR**1654068**, 10.1090/S0002-9939-99-05266-1

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47A30,
47A63,
47B15

Retrieve articles in all journals with MSC (2000): 47A30, 47A63, 47B15

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:
http://dx.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