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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
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Abstract: We show Schwarz type inequalities and consider their converses. A continuous function $f : [0, \infty) \rightarrow [0, \infty)$ is said to be semi-operator monotone on $(a,b)$ if $ \{f( t^{\frac{1}{2}} ) \}^{2}$ is operator monotone on $(a^{2},b^{2})$. Let $T$ be a bounded linear operator on a complex Hilbert space ${\mathcal H}$ and $ T = U \vert T \vert $ be the polar decomposition of $ T$. Let $ 0 \leq A, B \in B( {\mathcal H})$ and $ \Vert Tx \Vert \leq \Vert Ax\Vert, \Vert T^{*} y \Vert \leq \Vert By \Vert $ for $ x, y \in {\mathcal H}$. (1) If a non-zero function $f$ is semi-operator monotone on $(0, \infty)$, then $ \vert \langle Tx, y \rangle \vert \leq \Vert f(A) x \Vert \Vert g(B) y \Vert $ for $ x, y \in {\mathcal H}$, where $ g(t) = t/f(t)$. (2) If $f, g$ are semi-operator monotone on $(0, \infty)$, then $ \vert \langle U f(\vert T \vert)g(\vert T \vert)x, y \rangle \vert \leq \Vert f(A) x \Vert \Vert g(B) y \Vert $ for $ x, y \in {\mathcal H}$. 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

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

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

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

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