Abstract 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 of ball-like free 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.

中文翻译：

---

谱面体和谱球之间的双分析自由图

摘要 线性矩阵不等式 (LMI) 在实代数几何、半定规划、控制理论和信号处理中无处不在。具有（无维度）矩阵未知数的 LMI 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。LMI 的矩阵可行性集称为自由谱面体。在本文中，对一类非常一般的球状自由谱面体（其示例包括行或列收缩以及收缩元组）和任意自由谱面体之间的二元分析映射进行了表征，并认为它们具有优雅的代数形式。它们都是高度结构化的理性映射。在域和codomain都是球状的情况下，这些二元分析图是明确确定的，文章给出了存在这样一个具有指定值和点导数的地图的充分必要条件。特别是，这个结果导致了球状自由谱面体的自同构群的分类。证明依赖于一种新颖的自由 Nullstellensatz，只有在开发和应用自由分析中的新工具以获取关于球状自由谱面体边界的精细细节、局部几何性质和全局代数性质后才建立。