Asymptotically Optimal Multi-Paving
Asymptotically Optimal Multi-Paving
复制标题
渐近最优多重铺路
DOI：
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发表时间：
2017
影响因子：
1
通讯作者：
N. Srivastava
中科院分区：
文献类型：
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作者：
M. Ravichandran;N. Srivastava
Anderson的铺装猜想，现在被认为是正确的，它断言每一个零对角线矩阵都允许一个具有维数无关边界的非平凡铺装。我们研究了一个矩阵集合的这一问题，证明了给定$k$ 0对角线$n乘以n$赫米矩阵$A_1,ldots,A_k$和$epsilon> $，存在$P_1,ldots,P_r$的对角投影$P_1,ldots,P_r$与$sum _{jle r} P_j=I$，使得$||P_{j}A_iP_{j}||le epsilon ||A_i||$对于$ile k, jle r$，即同时铺装这些矩阵，$rle 18k/epsilon ^2$。因此，我们得到了铺成单个零对角（不一定是厄米）矩阵的最优渐近估计：每个平方零对角复矩阵都可以用$O(epsilon ^{-2})$块铺成$epsilon -$，改进了先前已知的$O(epsilon ^{-8})$的界。我们用我们的结果加强了Johnson-Ozawa-Schechtman关于零迹矩阵的对易子表示的一个结果。
Anderson’s paving conjecture, now known to be true [14], asserts that every zero-diagonal matrix admits a nontrivial paving with dimension independent bounds. We study this problem for a collection of matrices and show that given $k$ zero-diagonal $n imes n$ Hermitian matrices $A_1,ldots ,A_k$ and $epsilon>0$ there are diagonal projections $P_1,ldots ,P_r$ with $sum _{jle r} P_j=I$ such that $||P_{j}A_iP_{j}||le epsilon ||A_i||$ for $ile k, jle r$, that is, a simultaneous paving of the matrices, with $rle 18k/epsilon ^2$. As a consequence, we get the optimal asymptotic estimates for paving a single zero-diagonal (not necessarily Hermitian) matrix: every square zero-diagonal complex matrix can be $epsilon -$paved using $O(epsilon ^{-2})$ blocks, improving the previously known bound of $O(epsilon ^{-8})$. We use our result to strengthen a result of Johnson–Ozawa–Schechtman on commutator representations of zero trace matrices.