$H^\infty$迹代数的可逆性阈值和有效矩阵求逆
Invertibility threshold for $H^\infty$ trace algebras, and effective matrix inversions
摘要：
对于一个给定的$δ$，$0<δ<1$，构造一个Blaschke序列$σ=\{λ_j\}$，使得具有$δ<δ_f=\inf_{λ\inσ}|f(λ)|\le\|f\|_\infty\le1$的每个函数$f$，$f\in H^\infty$在迹代数$H^\infty|σ$中是可逆的（其逆的范数估计仅取决于$δ_f$），但存在具有$δ=δ_f\le\|f\|_\infty\le1$的$f$，它不可逆。作为应用，给出了有界算子的Bourgain-Tzafriri限制可逆性猜想的一个反例，其中一个“正交（或无条件）基”被一个"求和块正交基“所取代.
英文摘要：
For a given $δ$, $0<δ<1$, a Blaschke sequence $σ=\{λ_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $δ<δ_f=\inf_{λ\inσ}|f(λ)|\le\|f\|_\infty\le1$ is invertible in the trace algebra $H^\infty|σ$ (with a norm estimate of the inverse depending on $δ_f$ only), but there exists $f$ with $δ=δ_f\le\|f\|_\infty\le1$, 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''.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
