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Matrix inequalities and majorizations around Hermite-Hadamard's inequality

Abstract : We study the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates ∥A q ∥ p p + ∥B q ∥ p p 1/p ≤ ∥(xA + (1 − x)B)) q ∥ p + ∥(1 − x)A + xB) q ∥ p for all positive (semidefinite) n × n matrices A, B and 0 < q, x < 1. A related decomposition, with the assumption X * X + Y * Y = XX * + Y Y * = I, is (X * AX + Y * BY) ⊕ (Y * AY + X * BX) = 1 2n 2n k=1 U k (A ⊕ B)U * k for some family of 2n × 2n unitary matrices U k. This is a majorization which is obtained by using the Hansen-Pedersen trace inequality.
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https://hal.archives-ouvertes.fr/hal-03526745
Contributor : Jean-Christophe Bourin Connect in order to contact the contributor
Submitted on : Friday, January 14, 2022 - 4:43:23 PM
Last modification on : Friday, April 22, 2022 - 2:22:03 PM
Long-term archiving on: : Friday, April 15, 2022 - 7:12:53 PM

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Jean-Christophe Bourin, Eun-Young Lee. Matrix inequalities and majorizations around Hermite-Hadamard's inequality. Canadian Mathematical Bulletin, Cambridge University Press, inPress. ⟨hal-03526745⟩

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