论文标题
多盘和通勤收缩的扩张中的杰出品种
Distinguished varieties in the polydisc and dilation of commuting contractions
论文作者
论文摘要
polydisc $ \ mathbb d^n $的杰出品种是一种仿生的代数变体，与$ \ mathbb d^n $相交，并通过$ n $ -torus $ \ mathbb t^n $离开域，而无需相交的任何其他部分$ \ mathbb d^n $的拓扑边界。我们在某些线性矩阵铅笔的Taylor关节频谱上找到了在Polydisc $ \ Mathbb d^n $中具有杰出品种的两种不同特征，从而概括了由于Agler和M rish.45ex \ Hbox {C} Carthy [acta carthy [Acta Math。我们表明，$ \ mathbb d^n $中的杰出品种是仿射代数曲线的一部分，这是一个固定的理论完整交叉点。 We also show that if $(T_1, \dots , T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a commuting unitary dilation $(U_1, \dots , U_n)$ with $ u = \ prod_ {i = 1}^n u_i $是$ t $的最小统一扩张，并且仅当某些与$（t_1，\ dots，t_n）$相关的某些某些矩阵定义了$ \ mathbb d^n $中的杰出品种。
A distinguished variety in the polydisc $\mathbb D^n$ is an affine complex algebraic variety that intersects $\mathbb D^n$ and exits the domain through the $n$-torus $\mathbb T^n$ without intersecting any other part of the topological boundary of $\mathbb D^n$. We find two different characterizations for a distinguished variety in the polydisc $\mathbb D^n$ in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and M\raise.45ex\hbox{c}Carthy [Acta Math., 2005] on distinguished varieties in $\mathbb D^2$. We show that a distinguished variety in $\mathbb D^n$ is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if $(T_1, \dots , T_n)$ is commuting tuple of Hilbert space contractions such that the defect space of $T=\prod_{i=1}^n T_i$ is finite dimensional, then $(T_1, \dots , T_n)$ admits a commuting unitary dilation $(U_1, \dots , U_n)$ with $U=\prod_{i=1}^n U_i$ being the minimal unitary dilation of $T$ if and only if some certain matrices associated with $(T_1, \dots , T_n)$ define a distinguished variety in $\mathbb D^n$.
