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

Abstract:

Given $\delta$, $0<\delta <1$, a Blaschke sequence $\sigma =\{\lambda _j\}$ is constructed such that every function $f\in H^\infty$ satisfying $\delta <\delta _f=\inf _{\lambda \in \sigma }|f(\lambda )|\le \|f\|_\infty \le 1$ 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 $\delta =\delta _f\le \|f\|_\infty \le 1$ that is 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”.