Extreme Points of Matrix Convex Sets, Free Spectrahedra, and Dilation Theory
Extreme Points of Matrix Convex Sets, Free Spectrahedra, and Dilation Theory
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矩阵凸集的极值点、自由谱面和膨胀理论
DOI：
10.1007/s12220-017-9866-4
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
S. McCullough
中科院分区：
文献类型：
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作者：
Eric Evert;J. Helton;I. Klep;S. McCullough
对于矩阵凸集，给出了极值点和Arveson边界点概念的统一几何解释。这些概念包括，按照强度递增的顺序，“欧几里得”极值点，“矩阵”极值点和“绝对”极值点的核心概念。一个看似不同的概念，“Arveson边界”，相比之下，具有扩张理论的味道。Arveson边界点是算子系统的边界表示（不一定是不可约的）的模拟。这篇文章提供并探索了上述极值点概念的膨胀理论公式。线性矩阵不等式（LMI）的标量解集称为谱面体。线性矩阵不等式的矩阵解集是一个自由谱面体。光谱面体（分别为自由谱面体）位于一般凸集（分别为矩阵凸集）和凸多面体（分别为自由多面体）。作为极值点定理的应用，证明了矩阵凸集K的极对偶是由n个Arveson边界点生成的当且仅当K是自由谱面体，并且如果自由谱面体K的极对偶也是自由谱面体，则K在标量水平上是多面体.
For matrix convex sets, a unified geometric interpretation of notions of extreme points and of Arveson boundary points is given. These notions include, in increasing order of strength, the core notions of “Euclidean” extreme points, “matrix” extreme points, and “absolute” extreme points. A seemingly different notion, the “Arveson boundary”, has by contrast a dilation-theoretic flavor. An Arveson boundary point is an analog of a (not necessarily irreducible) boundary representation for an operator system. This article provides and explores dilation-theoretic formulations for the above notions of extreme points. The scalar solution set of a linear matrix inequality (LMI) is known as a spectrahedron. The matricial solution set of an LMI is a free spectrahedron. Spectrahedra (resp. free spectrahedra) lie between general convex sets (resp. matrix convex sets) and convex polyhedra (resp. free polyhedra). As applications of our theorems on extreme points, it is shown that the polar dual of a matrix convex set K is generated, as a matrix convex set, by finitely many Arveson boundary points if and only if K is a free spectrahedron, and if the polar dual of a free spectrahedron K is again a free spectrahedron, then at the scalar level K is a polyhedron.
多个非交换变量的全局全纯函数 II
DOI：
10.4153/cmb-2017-044-4
发表时间：
2018
期刊：
Canadian Mathematical Bulletin
影响因子：
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作者：
Agler, Jim;McCarthy, John
通讯作者：
McCarthy, John