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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Further extension
of the Heinz-Kato-Furuta inequality


Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 127 (1999), 2899-2904
MSC (1991): Primary 47A63, 47A30, 47B25
Published electronically: April 23, 1999
MathSciNet review: 1654068
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Abstract: Let $T$ be a bounded operator on a Hilbert space $\mathfrak{H},$ and $A,B$ positive definite operators. Kato has shown that if $\Vert T x \Vert \leq \Vert A x \Vert $ and $ \Vert T^{*} y \Vert \leq \Vert B y \Vert $ for all $x, y \in \mathfrak{H}$, then $|(Tx , y)| \leq ||f(A)x|| \; ||g(B)y||, $ where $f(t), \ g(t)$ are operator monotone functions defined on $[0, \infty )$ such that $f(t) g(t)=t$. Furuta has shown that $|(T|T|^{\alpha +\beta -1}x,y)|\leq ||A^{\alpha }x|| \ ||B^{\beta }y||, \text{ where } 0 \leq \alpha ,\beta \leq 1, \ 1\leq \alpha + \beta .$ Let $f(t), \ g(t)$ be any continuous operator monotone functions, and set $h(t) = f(t) g(t) /t$ for $t >0.$ We will show that $Th(|T|)$ is well defined and $|(T h(|T|)x,y)| \leq ||f(A)x|| \; ||g(B)y||.$ Moreover, we will extend this result for unbounded closed operators densely defined on $\mathfrak{H}.$


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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka, 811-4192, Japan
Email: uchiyama@fukuoka-edu.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05266-1
PII: S 0002-9939(99)05266-1
Keywords: L\"{o}wner theory, Heinz inequality, operator monotone function, closed operator, non-negative operator
Received by editor(s): June 6, 1997
Received by editor(s) in revised form: December 9, 1997
Published electronically: April 23, 1999
Additional Notes: This research was partially supported by Grant-in-Aid for Scientific Research.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society