一致有界矩阵的随机铺路性质
The random paving property for uniformly bounded matrices
摘要：
本文对Bourgain和Tzafriri的一个重要结果给出了新的证明，该结果部分地解决了Kadison-Singer问题。结果表明，对于每个元素与维数相比相对较小的单位范数矩阵，都可以用一个定长的剖分来铺成。也就是说，坐标可以被划分成恒定数量的块，使得矩阵对每个坐标块的限制具有小于小于一半的范数。Bourgain和Tzafriri的原始证明涉及一个漫长而微妙的计算。新的证明依赖于对称化和（非交换）欣钦不等式的系统使用来估计一些随机矩阵的范数。
英文摘要：
This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and (noncommutative) Khintchine inequalities to estimate the norms of some random matrices.
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