Random sets of isomorphism of linear operators on Hilbert space

Details Random sets of isomorphism of linear operators on Hilbert space Random sets of isomorphism of linear operators on Hilbert space

2006-01-06

arxiv.org/abs/math/0601112v3

IMS Lecture Notes Monograph Series 2006, Vol. 51, 148-154

Mathematics

Functional Analysis

Published at http://dx.doi.org/10.1214/074921706000000815 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921706000000815

This note deals with a problem of the probabilistic Ramsey theory in functional analysis. Given a linear operator $T$ on a Hilbert space with an orthogonal basis, we define the isomorphic structure $\Sigma(T)$ as the family of all subsets of the basis so that $T$ restricted to their span is a nice isomorphism. Our main result is a dimension-free optimal estimate of the size of $\Sigma(T)$. It improves and extends in several ways the principle of restricted invertibility due to Bourgain and Tzafriri. With an appropriate notion of randomness, we obtain a randomized principle of restricted invertibility.

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