A characterization of idempotent Schur multipliers
Abstract
We prove that every idempotent Schur multiplier is a finite signed sum of contractive idempotent Schur multipliers. This was conjectured by Katavolos and Paulsen in 2003 and previously known only for translation-invariant Schur multipliers, by the Cohen-Host idempotent theorem. Concretely, we show that any boolean matrix A with Schur multiplier norm at most γ (or equivalently lVert ArVertγ2 ≤ γ) can be written as A=∑i=1Lσi Bi, where L≤ 2Cγ6 for an absolute constant C, σi∈{−1,1} are signs, and each Bi is a contractive idempotent Schur multiplier, that is, a boolean matrix whose $1$-entries form a union of all-one rectangular blocks, with no two blocks sharing a row or a column. As observed by Carenini, a key lemma in our work yields a new proof of the Cohen-Host theorem and gives a simple proof of the quantitative refinements of Green-Sanders and Sanders, with improved bounds. We include a self-contained exposition of these results in the case of finite groups.
Related
- Finite abelian groups are Schur multipliers of finite solvable groups
- Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem
- Characterisations for coarse embeddability via quasi-locality and Schur multipliers
- Schur Multipliers with Unequal Operator and Completely Bounded Norms on Sp, $1<p\ne 2<\infty $
- Optimal square size for separating the operator and completely bounded norms of Schur multipliers on S^4
BibTeX
@misc{beke2026characterization,
title = {{A characterization of idempotent Schur multipliers}},
author = {Csongor Beke and Marcel K. Goh and Hamed Hatami and Sean Jaffe and Daniel Naylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.14316},
eprint = {2607.14316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14316},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A characterization of idempotent Schur multipliers
AU - Csongor Beke
AU - Marcel K. Goh
AU - Hamed Hatami
AU - Sean Jaffe
AU - Daniel Naylor
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2607.14316
UR - https://arxiv.org/abs/2607.14316
AB - We prove that every idempotent Schur multiplier is a finite signed sum of contractive idempotent Schur multipliers. This was conjectured by Katavolos and Paulsen in 2003 and previously known only for translation-invariant Schur multipliers, by the Cohen-Host idempotent theorem. Concretely, we show that any boolean matrix A with Schur multiplier norm at most γ (or equivalently lVert ArVert_γ_2 ≤ γ) can be written as A=∑_i=1^Lσ_i B_i, where L≤ 2^Cγ^6 for an absolute constant C, σ_i∈{−1,1} are signs, and each B_i is a contractive idempotent Schur multiplier, that is, a boolean matrix whose $1$-entries form a union of all-one rectangular blocks, with no two blocks sharing a row or a column. As observed by Carenini, a key lemma in our work yields a new proof of the Cohen-Host theorem and gives a simple proof of the quantitative refinements of Green-Sanders and Sanders, with improved bounds. We include a self-contained exposition of these results in the case of finite groups.
ER -