This is an announcement for the paper "A new proof of the paving property for uniformly bounded matrices" by Joel A. Tropp.
Abstract: 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 Khintchine inequalities to estimate the norm of some random matrices. The key new ideas are due to Rudelson.
Archive classification: Metric Geometry; Functional Analysis; Probability
Mathematics Subject Classification: 46B07; 47A11; 15A52
Remarks: 12 pages
The source file(s), bdd-ks-v1.bbl: 2693 bytes, bdd-ks-v1.tex: 41646 bytes, macro-file.tex: 8551 bytes, is(are) stored in gzipped form as 0612070.tar.gz with size 15kb. The corresponding postcript file has gzipped size 99kb.
Submitted from: jtropp@umich.edu
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http://front.math.ucdavis.edu/math.MG/0612070
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http://arXiv.org/abs/math.MG/0612070
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