bibkey: zhang2024clarksonmccarthy
authors: Teng Zhang
year: 2024
title: "Clarkson–McCarthy type inequalities, part I: ℓ_p–ℓ_p and ℓ_q–ℓ_p Schatten p-estimates"
doi: null
url: https://arxiv.org/abs/2410.21961v4
claim: "The paper proves Clarkson–McCarthy type inequalities for Schatten p-norms, including the sharp Euler-type estimate with constant 2^(p-2) for three operators, and conjectures, from numerical evidence, the sharp three-operator inequality ||A+B+C||_p^p+||A||_p^p+||B||_p^p+||C||_p^p <= (3^(p-1)+1)/2^p (||A+B||_p^p+||B+C||_p^p+||C+A||_p^p) for p >= 2, reversed for 0 < p <= 2."
strata_touched:
  - D5/S3/Quantum/Matrix/ZhangThreeOperatorClarksonRefutation
license: citation-only
triage: anchor

Zhang, Clarkson–McCarthy type inequalities, part I

T. Zhang, arXiv:2410.21961v4 [math.FA] (28 March 2026; v1 29 October 2024). Quotations are from the v4 TeX source, with the macro \tr written as $\mathrm{tr}$. The same conjecture is restated in T. Zhang, arXiv:2602.01463v2.

The Schatten norm:

For $1\le p<\infty$, let $S_p$ denote the Schatten–von Neumann class of compact operators $A:{\ell }_2\to {\ell }_2$ endowed with the norm $\Vert A{\Vert }_p=(\mathrm{tr}(A^{\ast }A{)}^{\frac{p}{2}}{)}^{\frac{1}{p}}=({\sum }_j{s}_j(A{)}^p{)}^{\frac{1}{p}}$

The conjecture, introduced by “numerical evidence suggests the following sharp inequality”:

Let $A,B,C\in S_p$. Then for $2\le p<\infty$, $$\Vert A+B+C{\Vert }_p^p+\Vert A{\Vert }_p^p+\Vert B{\Vert }_p^p+\Vert C{\Vert }_p^p\le \frac{3^{p-1}+1}{2^p}\left(\Vert A+B{\Vert }_p^p+\Vert B+C{\Vert }_p^p+\Vert C+A{\Vert }_p^p\right).$$ For $0<p\le 2$, the inequality is reversed. Moreover, the constant $\frac{3^{p-1}+1}{2^p}$ is optimal: equality holds for all $p>0$ by taking $A=B=C\ne 0$.

Verified locator

Source: https://arxiv.org/abs/2410.21961v4

• URL: https://arxiv.org/abs/2410.21961v4 (the e-print is a gzipped tar archive; member v1_Clarkson_McCarthy_type_inequalities_part_I.tex, 2399 lines, md5 3ba34aa0913b51484593b3594a9f69d6):
  • the Schatten norm, Introduction, l. 100–104;
  • the conjecture, conj:sharp-inequality, l. 571–581.
• The restatement: arXiv:2602.01463v2, member On_five_questions_of_Bourin_and_Lee.tex (2713 lines, md5 b3e91436b76494a53687b26bf5b2d06e), l. 411–419.
• The author and the version history are from the arXiv abstract page; no journal reference is listed.