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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 

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