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ISSN 1088-6826(e) ISSN 0002-9939(p)

Inequalities of Reid type and Furuta

Author(s): C.-S. Lin
Journal: Proc. Amer. Math. Soc. 129 (2001), 855-859.
MSC (1991): Primary 47A63
Posted: September 20, 2000
MathSciNet review: 1709759
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Abstract:

Two of the most useful inequality formulas for bounded linear operators on a Hilbert space are the Löwner-Heinz and Reid's inequalities. The first inequality was generalized by Furuta (so called the Furuta inequality in the literature). We shall generalize the second one and obtain its related results. It is shown that these two generalized fundamental inequalities are all equivalent to one another.

References:

[1]: T. Furuta, A simplified proof of Heinz inequality and scrutiny of its equality, Proc. Amer. Math. Soc., 97(1986), 751-753. MR 87h:47016
[2]: T. Furuta. $A\geq B\geq O$ assures $(B^rA^pB^r)^{1/q}\geq B^{(p+2r)/q}$ for $r\geq 0,$ $p\geq 0,$ $q\geq 1$ with $(1+2r)q\geq p+2r,$ Proc. Amer. Math. Soc., 101(1987), 85-88. MR 89b:47028
[3]: T. Furuta, Equivalence relations among Reid, Löwner-Heinz and Heinz-Kato inequalities, and extensions of these inequalities, Integr. Equ. Oper. Theory, 29(1997), 1-9. MR 98f:47022
[4]: P. R. Halmos, Hilbert Space Problem Book, Van Nostrand, Princeton, N. J. 1967. MR 34:8178
[5]: C.-S. Lin, On Heinz-Kato type characterizations of Furuta inequality, Nihonkai Math. J., 9(1998), 187-191. MR 99j:47021
[6]: C.-S. Lin, On Halmos' sharpening of Reid's inequality, C. R. Math. Rep. Acad. Sci. Canada, 20(1998), 62-64. MR 99a:47015
[7]: K. Löwner, Über monotone Matrixfunktionen, Math. Z., 38(1934), 177-216.
[8]: W. T. Reid, Symmetrizable completely continuous linear transformations in Hilbert space, Duke Math. J., 18(1951), 41-56. MR 13:564b

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Additional Information:

C.-S. Lin
Affiliation: Department of Mathematics, Bishop's University, Lennoxville, Quebec, Canada J1M 1Z7
Email: plin@ubishops.ca

DOI: 10.1090/S0002-9939-00-05650-1
PII: S 0002-9939(00)05650-1
Keywords: Positive operator, Hermitian operator, polar decomposition, L\"{o}wner-Heinz inequality, Reid's inequality, Furuta's inequality, contraction
Received by editor(s): May 25, 1999
Posted: September 20, 2000
Dedicated: Dedicated to Professor Jone Lin on his retirement
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2000, American Mathematical Society

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