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A Converse of the Loewner–Heinz Inequality, Geometric Mean and Spectral Order

Published online by Cambridge University Press:  13 March 2014

Mitsuru Uchiyama
Mitsuru Uchiyama*
Affiliation:
Department of Mathematics, Interdisciplinary Faculty of Science and Engineering, Shimane University, Matsue City, Shimane 690-8504, Japan, (uchiyama@riko.shimane-u.ac.jp)
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Abstract

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Let A, B be non-negative bounded self-adjoint operators, and let a be a real number such that 0 < a < 1. The Loewner–Heinz inequality means that A ≤ B implies that Aa ≦ Ba. We show that A ≤ B if and only if (A + λ)a ≦ (B + λ)a for every λ > 0. We then apply this to the geometric mean and spectral order.

Keywords

Loewner–Heinz inequalitygeometric meanspectral orderPrimary 47A6347A64Secondary 15A3947A60
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 565 - 571
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Bhatia, R., Positive definite matrices, Princeton Series in Applied Mathematics (Princeton University Press, 2007).Google Scholar
2.Koranyi, A., On a theorem of Löwner and its connections with resolvents of self-adjoint transformations, Acta Sci. Math. (Szeged) 17 (1956), 63–70.Google Scholar
3.Kubo, F. and Ando, T., Means of positive linear operators, Math. Annalen 246 (1980), 205–224.Google Scholar
4.Löwner, K., Über monotone Matrix functionen, Math. Z. 38 (1934), 177–216.Google Scholar
5.Olson, M. P., The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. Am. Math. Soc. 28 (1971), 537–544.Google Scholar
6.Uchiyama, M., Commutativity of self-adjoint operators, Pac. J. Math. 161 (1993), 385–392.CrossRefGoogle Scholar
7.Uchiyama, M., A new majorization between functions, polynomials, and operator inequalities, J. Funct. Analysis 231 (2006), 221–244.Google Scholar