The random paving property for uniformly bounded matrices

Joel A. Tropp¹

¹ Applied & Computational Mathematics, MC 217-50 California Institute of Technology 1200 E. California Blvd. Pasadena, CA 91125-5000, U.S.A.

Studia Mathematica, Tome 185 (2008) no. 1, pp. 67-82

Résumé

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.

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DOI : 10.4064/sm185-1-4

Keywords: note presents proof important result due bourgain tzafriri provides partial solution kadison singer problem result shows every unit norm matrix whose entries relatively small comparison its dimension paved partition constant size coordinates partitioned constant number blocks restriction matrix each block coordinates has norm half original proof bourgain tzafriri involves long delicate calculation proof relies systematic symmetrization noncommutative khinchin inequalities estimate norms random matrices