Skip to content
DraconiticResearch
Search Sources API
190,335 studies31 sourcesUpdated 37 min ago
PreprintarXiv licencearXiv

Schur bounded patterns, submajorisation and operator Lipschitz functions

  • Edward McDonald
  • Fedor Sukochev
  • Dmitriy Zanin
Open at sourcePDFDOICite

Abstract

A Schur bounded pattern is a subset S⊂ ℕ2 such that Schur multiplication by every bounded function on ℕ2 supported on S defines a bounded linear operator in the norm of ℬ(ℓ2(ℕ)). Schur bounded patterns were characterised by Davidson-Donsig as being unions of row-bounded and column-bounded sets. We study the analogous question for sets S such that element-wise multiplication by every bounded function on S is bounded in ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals ℒp with 0<p<1 and the weak Schatten ideal ℒ1,∞. Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns. This has implications for the functions which are Lipschitz in the norm of ideals that are not closed under submajorisation. In particular such functions must be differentiable and have derivative that is asymptotically constant at infinity.

Related

  1. Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below
    Yin Jiang, Nan Li
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  2. Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain
    Zhongwei Shen
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
    Wenxian Ma, Sibei Yang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. A Separable Hilbertian Operator Space with CBAP and No Completely Bounded Basis
    Dominique Guillot, Nicolas Cutrona
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  5. Lpμ-Normed Spaces of Bounded Multilinear Operator
    Filippo de Feo
    arXivPreprintCC0Full textAnalysis & PDEsProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{mcdonald2026schur,
  title = {{Schur bounded patterns, submajorisation and operator Lipschitz functions}},
  author = {Edward McDonald and Fedor Sukochev and Dmitriy Zanin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.07564},
  eprint = {2605.07564},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.07564},
}
RIS
TY  - UNPB
TI  - Schur bounded patterns, submajorisation and operator Lipschitz functions
AU  - Edward McDonald
AU  - Fedor Sukochev
AU  - Dmitriy Zanin
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2605.07564
UR  - https://arxiv.org/abs/2605.07564
AB  - A Schur bounded pattern is a subset S⊂ ℕ^2 such that Schur multiplication by every bounded function on ℕ^2 supported on S defines a bounded linear operator in the norm of ℬ(ℓ_2(ℕ)). Schur bounded patterns were characterised by Davidson-Donsig as being unions of row-bounded and column-bounded sets. We study the analogous question for sets S such that element-wise multiplication by every bounded function on S is bounded in ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals ℒ_p with 0<p<1 and the weak Schatten ideal ℒ_1,∞. Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns. This has implications for the functions which are Lipschitz in the norm of ideals that are not closed under submajorisation. In particular such functions must be differentiable and have derivative that is asymptotically constant at infinity.
ER  -