AI 中文总结
研究幂等舒尔乘子，推测其可写成有限个压缩幂等元的和，等同于特定布尔矩阵的表示形式。证明满足条件的布尔矩阵可按特定形式表示，且给出$L$的上界。应用此结论得出分解范数有界的矩阵序列所属复杂度类。
AI 中文摘要
据推测，每个幂等舒尔乘子都可写成有限个压缩幂等元的和。该推测等同于：任何分解范数$\lVert A\rVert_{\gamma_2}$至多为$\gamma$的布尔矩阵$A$，可表示为符号和$A = \sum_{i=1}^L \pm B_i$，其中经行列置换后，每个$B_i$是单位矩阵的放大，且$L$仅取决于$\gamma$。本文表明，若$n\times n$布尔矩阵$A$满足$\lVert A\rVert_{\gamma_2} \le \gamma$，则它可表示为$L = 2^{O(\gamma^9) + \log^*\! n}$的上述形式，$\log^*$是迭代对数函数。作为应用，任何分解范数有界的矩阵序列属于通信问题的复杂度类$\mathrm{P}^\mathrm{EQ}$，其具有多对数等式预言机复杂度。
英文摘要
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. 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.
Comments15 pages, including references
