Improvement on a Generalized Lieb's Concavity Theorem

2019/05/04 by De Huang, Huang, De
Computer Science · Mathematics · #15A16 #15A42 #47A57 #47A63 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Matrix Theory and Algorithms #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1905.02194

openalex publication_date 2019/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that Lieb's concavity theorem holds more generally for any unitary invariant matrix function ϕ:H₊ⁿ→ ℝ₊ⁿ that is concave and satisfies Hölder's inequality. Concretely, we prove the joint concavity of the function (A,B) ↦ϕ[(B^(qs)/(2)K^*A^psKB^(qs)/(2))^(1)/(s)] on H₊ⁿ\timesH₊^m, for any K∈ ℂ^{n× m} and any s,p,q∈(0,1], p+q≤ 1. This result improves a recent work by Huang for a more specific class of ϕ.

Citations

• Interpolation of linear operators

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