This is an announcement for the paper "The infinite dimensional restricted invertibility theorem" by Peter G. Casazza and Goetz E. Pfander.

Abstract: The 1987 Bourgain-Tzafriri Restricted Invertibility Theorem is one of the most celebrated theorems in analysis. At the time of their work, the authors raised the question of a possible infinite dimensional version of the theorem. In this paper, we will give a quite general definition of restricted invertibility for operators on infinite dimensional Hilbert spaces based on the notion of "density" from frame theory. We then prove that localized Bessel systems have large subsets which are Riesz basic sequences. As a consequence, we prove the strongest possible form of the infinite dimensional restricted invertibility theorem for $\ell_1$-localized operators and for Gabor frames with generating function in the Feichtinger Algebra. For our calculations, we introduce a new notion of "density" which has serious advantages over the standard form because it is independent of index maps - and hence has much broader application. We then show that in the setting of the restricted invertibility theorem, this new density becomes equivalent to the standard density.

Archive classification: math.FA math.CA

Mathematics Subject Classification: 42C15, 46C05, 46C07

Remarks: 24 pages

The source file(s), PaperArxiv.tex: 85007 bytes, is(are) stored in gzipped form as 0905.0656.gz with size 24kb. The corresponding postcript file has gzipped size 143kb.

Submitted from: g.pfander@jacobs-university.de

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