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For a given $\delta$, $0<\delta<1$, a Blaschke sequence$\sigma=\{\lambda_j\}$ is constructed such that every function $f$, $f\inH^\infty$, having$\delta<\delta_f=\inf_{\lambda\in\sigma}|f(\lambda)|\le\|f\|_\infty\le1$ isinvertible in the trace algebra $H^\infty|\sigma$ (with a norm estimate of theinverse depending on $\delta_f$ only), but there exists $f$ with$\delta=\delta_f\le\|f\|_\infty\le1$, which does not. As an application, acounterexample to a stronger form of the Bourgain--Tzafriri restrictedinvertibility conjecture for bounded operators is exhibited, where an``orthogonal (or unconditional) basis'' is replaced by a ``summation blockorthogonal basis''.

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