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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
Abstract
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.
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- Copyright © Edinburgh Mathematical Society 2014
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