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) Khinchin inequalities to estimate the norms of some random matrices.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-1-4,
author = {Joel A. Tropp},
title = {The random paving property for uniformly bounded matrices},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {67-82},
zbl = {1152.46007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-1-4}
}
Joel A. Tropp. The random paving property for uniformly bounded matrices. Studia Mathematica, Tome 187 (2008) pp. 67-82. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-1-4/