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VOL. 51 | 2006 Random sets of isomorphism of linear operators on Hilbert space


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.


Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1129.46006
MathSciNet: MR2387766

Digital Object Identifier: 10.1214/074921706000000815

Primary: 46B09 , 47D25

Keywords: operators on Hilbert spaces , restricted invertibility

Rights: Copyright © 2006, Institute of Mathematical Statistics


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