论文标题
三角形四足动，收缩分解，扩张和亚变量
Triangular Tetrablock-contractions, factorization of contractions, dilation and subvarieties
论文作者
论文摘要
Hilbert太空运营商$（a，b，p）的通勤三重$，封闭的四载四元$ \ bar {\ mathbb e} $是一个频谱集，称为a \ textIt {tetrableock-contraction}或简单地称为$ \ mathbb e $ - \ textit { e = \ {（a_ {11}，a_ {22}，\ det a）：\，a = [a_ {ij}] \ in \ mathcal m_2（\ mathbb c），\; \ | a \ | <1 \} \ subset \ mathbb c^3 \]是一个多项式凸面域，自然与$ $ $ - $ - 同伴问题相关联。我们介绍了三角形$ \ mathbb e $ - 承担，并证明每个纯三角$ \ mathbb e $ $ - 收集都会扩展到纯三角形$ \ mathbb e $ -isometry。我们为纯三角$ \ Mathbb e $ iSometry构建功能模型，并应用该模型为著名的Berger-Coburn-Lebow-Lebow模型定理找到通勤异构体的新证明。接下来，我们为Berger-Coburn-Lebow模型的更广泛的版本提供了替代证明，即由于DAS，Sarkar和Sarkar引起的纯收缩（\ textit {adv。Math。} 322（2017），186-200）。我们发现，在最小的扩张空间中存在$ \ mathbb e $ $单独的扩张，以$ \ mathbb e $单独扩张（a，b，p）$，并表明它等于它与缺损空间$ d_ d_ d_ d_ d_ p^p^p^p^p^*pinite nimention的$ \ Mathbb e $中的杰出品种相当。
A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\bar{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(a_{11},a_{22}, \det A):\, A=[a_{ij}]\in \mathcal M_2(\mathbb C), \; \|A\| <1 \} \subset \mathbb C^3 \] is a polynomially convex domain which is naturally associated with the $μ$-synthesis problem. We introduce triangular $\mathbb E$-contractions and prove that every pure triangular $\mathbb E$-contraction dilates to a pure triangular $\mathbb E$-isometry. We construct a functional model for a pure triangular $\mathbb E$-isometry and apply that model to find a new proof for the famous Berger-Coburn-Lebow Model Theorem for commuting isometries. Next we give an alternative proof to the more generalized version of Berger-Coburn-Lebow Model, namely the factorization of a pure contraction due to Das, Sarkar and Sarkar (\textit{Adv. Math.} 322 (2017), 186 -- 200). We find a necessary and sufficient condition for the existence of $\mathbb E$-unitary dilation of an $\mathbb E$-contraction $(A,B,P)$ on the smallest dilation space and show that it is equivalent to the existence of a distinguished variety in $\mathbb E$ when the defect space $D_{P^*}$ is finite dimensional.
