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 $\lVert A\rVert_{γ_2} \le γ$) can be written as \[ A=\sum_{i=1}^{L}σ_i B_i,\] where $L\leq 2^{Cγ^6}$ for an absolute constant $C$, $σ_i\in\{-1,1\}$ 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.