Bianalytic free maps between spectrahedra and spectraballs
Bianalytic free maps between spectrahedra and spectraballs
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DOI：
10.1016/j.jfa.2020.108472
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发表时间：
2018-04
影响因子：
1.7
通讯作者：
J. Helton;I. Klep;S. McCullough;Jurij Volvcivc
中科院分区：
文献类型：
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作者：
J. Helton;I. Klep;S. McCullough;Jurij Volvcivc
线性矩阵不等式在真实的代数几何、半定规划、控制理论和信号处理等领域中普遍存在。具有（无维）矩阵未知数的线性矩阵不等式是完全正映射和算子代数、算子系统和空间理论的核心，并作为矩阵凸集的范式。线性矩阵不等式（LMI）的矩阵可行集称为自由谱面体，本文刻画了一类非常普遍的球状自由谱面体（包括行压缩、列压缩和压缩元组）与任意自由谱面体之间的双解析映射，并证明了其具有优美的代数形式。它们都是高度结构化的有理映射。在定义域和上定义域都是球形的情况下，本文明确地确定了这类双解析映射，并给出了这类映射存在的充要条件.特别是，这一结果导致了类球自由谱面体的自同构群的分类。证明依赖于一个新的自由零stellenstructures，建立后，只有在自由分析的新工具的开发和应用，以获得精细的细节，几何性质的本地和代数性质的全球性，关于边界的球状自由光谱面体。
Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron.In this article, the bianalytic maps between a very general class ofball-likefree spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this result leads to a classification of automorphism groups of ball-like free spectrahedra. The proofs depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra.