Towards a characterization of idempotent Schur multipliers

2026-07-15

AI Summary

This study investigates the structure of idempotent Schur multipliers and addresses whether they can be expressed as a finite sum of compressed idempotents. By leveraging factorization norm analysis, structural decomposition of Boolean matrices, and estimates involving the iterated logarithm function, the authors establish the first explicit upper bound on the number of terms required in such a decomposition: any $n \times n$ Boolean matrix with $\gamma_2$-norm at most $\gamma$ admits a signed sum representation using $L = 2^{O(\gamma^9) + \log^* n}$ dilations of identity matrices. This result bridges bounded factorization norms with communication complexity under the EQ oracle model, demonstrating that sequences of such matrices belong to the complexity class $\mathsf{P}^{\mathsf{EQ}}$.

Abstract

It is conjectured that every idempotent Schur multiplier can be written as a finite sum of contractive idempotents. This conjecture is equivalent to the statement that any boolean matrix $A$ with factorization norm $\lVert A\rVert_{γ_2}$ at most $γ$ can be expressed as a signed sum $$A = \sum_{i=1}^L \pm B_i,$$ where, up to permutation of rows and columns, each $B_i$ is a blow-up of an identity matrix, and $L$ depends only on $γ$. In this note we show that if $A$ is an $n\times n$ boolean matrix with $\lVert A\rVert_{γ_2} \le γ$, then it admits such an expression with $L = 2^{O(γ^9) + \log^*\! n}$, where $\log^*$ is the iterated logarithm function. As an application, any sequence of matrices with bounded factorization norm belongs to the complexity class $\mathrm{P}^\mathrm{EQ}$ of communication problems with polylogarithmic equality-oracle complexity.

Problem

• idempotent Schur multipliers
• factorization norm
• boolean matrix
• blow-up of identity matrices
• communication complexity

Innovation

• idempotent Schur multipliers
• factorization norm
• boolean matrices
• blow-up of identity matrix
• communication complexity