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 $\gamma$ (or equivalently $\Vert A{\Vert }_{{\gamma }_2}\le \gamma$) can be written as \[ A=\sum_{i=1}^{L}σ_i B_i,\] where $L\le 2^{C{\gamma }^6}$ for an absolute constant $C$, ${\sigma }_i\in \{-1,1\}$ 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.

15 pages, including references