Invertibility threshold for $H^{\infty }$ trace algebras, and effective matrix inversions

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 $\delta <{\delta }_f={\inf }_{\lambda \in \sigma }|f(\lambda )|\le \Vert f{\Vert }_{\infty }\le 1$ 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 $\delta ={\delta }_f\le \Vert f{\Vert }_{\infty }\le 1$, 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''.