This is an announcement for the paper "Invertibility threshold for $H^\infty$ trace algebras, and effective matrix inversions" by Nikolai Nikolski and Vasily Vasyunin.
Abstract: For a given $\delta$, $0<\delta<1$, a Blaschke sequence $\sigma={\lambda_j}$ is constructed such that every function $f$, $f\in H^\infty$, having $\delta<\delta_f=\inf_{\lambda\in\sigma}|f(\lambda)|\le|f|_\infty\le1$ is invertible in the trace algebra $H^\infty|\sigma$ (with a norm estimate of the inverse depending on $\delta_f$ only), but there exists $f$ with $\delta=\delta_f\le|f|_\infty\le1$, which does not. As an application, a counterexample to a stronger form of the Bourgain--Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an ``orthogonal (or unconditional) basis'' is replaced by a ``summation block orthogonal basis''.
Archive classification: math.FA
Submitted from: vasyunin@pdmi.ras.ru
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1010.6090
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