Coordinate Restrictions of Linear Operators in L

This paper addresses the problem of improving properties of a linear operator u in l n 2 by restricting it onto coordinate subspaces. We discuss how to reduce the norm of u by a random coordinate restriction, how to approximate u by a random operator with small " coordinate " rank, how to find coordinate subspaces where u is an isomorphism. The first problem in this list provides a probabilistic extension of a suppression theorem of B. Kashin and L. Tzafriri, the second one is a new look at a result of M. Rudelson on the random vectors in the isotropic position, the last one is the recent generalization of the Bourgain-Tzafriri's invertibility principle. The main point is that all the results are independent of n, the situation is instead controlled by the Hilbert-Schmidt norm of u. As an application, we provide an almost optimal solution to the problem of harmonic density in harmonic analysis, and a solution to the reconstruction problem for communication networks which deliver data with random losses.