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spectral order

Proceedings of the Edinburgh Mathematical Society (1)

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A Converse of the Loewner–Heinz Inequality, Geometric Mean and Spectral Order
Mitsuru Uchiyama
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 57 / Issue 2 / June 2014

Published online by Cambridge University Press:

13 March 2014, pp. 565-571
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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.