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

TL;DR

This paper constructs specific Blaschke sequences to analyze invertibility thresholds in $H^ty$ trace algebras, providing counterexamples to a strengthened invertibility conjecture for bounded operators.

Contribution

It introduces a novel construction of Blaschke sequences to determine invertibility thresholds and presents a counterexample to a stronger form of the Bourgain--Tzafriri conjecture.

Findings

Existence of Blaschke sequences with specific invertibility properties

Counterexample to a stronger invertibility conjecture

Norm estimates for inverses depending only on $\

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''.