Inequalities of Reid type and Furuta

Author:
C.-S. Lin

Journal:
Proc. Amer. Math. Soc. **129** (2001), 855-859

MSC (1991):
Primary 47A63

DOI:
https://doi.org/10.1090/S0002-9939-00-05650-1

Published electronically:
September 20, 2000

MathSciNet review:
1709759

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[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. assures for with 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:
https://doi.org/10.1090/S0002-9939-00-05650-1

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

Published electronically:
September 20, 2000

Dedicated:
Dedicated to Professor Jone Lin on his retirement

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2000
American Mathematical Society