标题： 零化理想和双圆盘中的Agler--McCarthy谱流形 显示英文标题

标题： Annhilating ideals and Agler--McCarthy spectral varieties in the bidisc

摘要： 闭单位双圆盘$\overline{\mathbb{D}}^2$被认为是任何一对交换压缩算子$(T_1,T_2)$的谱集。 当每个$T_i$为纯的且缺陷有限时，该对具有一个更小的谱集：位于双圆盘$\mathbb{D}^2$内的一个特殊流形$V$的闭包。 我们找到关于$(T_1,T_2)$的条件，以保证$V$的闭包是一个极小谱集。 此外，我们研究$V$与消去理想$\text{Ann}(T_1,T_2)$在$H^\infty(\mathbb{D}^2)$中的关系。 虽然$V$通常严格大于$\text{Ann}(T_1,T_2)$的零点集，但我们分离出一个自然的约束等距扩张$(S_1,S_2)$的$(T_1,T_2)$，其泰勒谱包含在$V$中，并与所谓的$\text{Ann}(T_1,T_2)$的支持密切相关。 我们还描述了当$\text{Ann}(T_1,T_2)$是在$(S_1^*,S_2^*)$的联合点谱上消失的函数的理想时的情况。 显示英文摘要

摘要： The closed unit bidisc $\overline{\mathbb{D}}^2$ is known to be a spectral set for any pair $(T_1,T_2)$ of commuting contractions. When each $T_i$ is pure and has finite defect, the pair admits a much smaller spectral set: the closure of a distinguished variety $V$ inside the bidisc $\mathbb{D}^2$. We find conditions on $(T_1,T_2)$ that guarantee that the closure of $V$ is a minimal spectral set. In addition, we examine the relationship between $V$ and the annihilating ideal $\text{Ann}(T_1,T_2)$ in $H^\infty(\mathbb{D}^2)$. While $V$ is typically strictly larger than the zero set of $\text{Ann}(T_1,T_2)$, we isolate a natural constrained isometric co-extension $(S_1,S_2)$ of $(T_1,T_2)$ whose Taylor spectrum is contained in $V$ and is closely linked to the so-called support of $\text{Ann}(T_1,T_2)$. We also characterize when $\text{Ann}(T_1,T_2)$ is the ideal of functions vanishing on the joint point spectrum of $(S_1^*,S_2^*)$.