Arveson极值点跨距自由谱面
Arveson extreme points span free spectrahedra
摘要：
令$ SM_n(\mathbb{R})^g$表示$n \times n$实对称矩阵的$g$ -元组.给定元组$X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$和$Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$，$X$和$Y$的矩阵凸组合是\[ V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n \]形式的和，其中$V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$和$V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$是收缩。矩阵凸集是在矩阵凸组合下闭的集合。矩阵凸组合的一个关键特征是$g$元组$X, Y$和$V_1^* XV_1+V_2^* Y V_2$不需要具有相同的大小。因此，矩阵凸集是凸集的无量纲类似物。虽然在经典的设置只有一个概念的极值点，有三个主要的概念极值点的矩阵凸集：普通，矩阵，和绝对极值点。绝对极值点与经典的Arveson边界密切相关。矩阵凸集理论的一个中心目标是确定这些类型的极值点之一是否通过矩阵凸组合最小地恢复了矩阵凸集。本文证明了由线性矩阵不等式定义的实紧矩阵凸集是其绝对极值点的矩阵凸包，绝对极值点是具有此性质的最小集。此外，我们给出了一个算法，它表示为一个绝对极值点的矩阵凸组合与最佳界的元组。在复数域而不是实数域上工作时，也有类似的结果。
英文摘要：
Let $ SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices. Given tuples $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form \[ V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n \] where $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and $V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the $g$-tuples $X, Y$, and $V_1^* XV_1+V_2^* Y V_2$ do not need to have the same size. As a result, matrix convex sets are a dimension free analog of convex sets. While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary. A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations. This article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are the minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals.
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。
