论文标题
自由非交通函数的整合性
Integrability of Free Noncommutative Functions
论文作者
论文摘要
非交通函数是在两个向量空间上的所有大小的平方矩阵之间的级函数，这些矢量空间尊重直接总和和相似性。它们具有非常强大的规律性特性（让人联想到通常的分析函数的规律性特性），并接受了良好的差异分辨率。非共同函数自然出现在各种环境中：非共同代数，系统和控制，光谱理论和自由概率。从J.L. Taylor的开创性工作开始，该理论是由D.-V.进一步开发的。 Voiculescu，近年来成为一个新的，非常活跃的研究领域。本文的目的是建立Frobenius集成性定理的非共同类似物：我们为具有抗激素的高级非交通函数提供了必要和足够的条件。
Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus. Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. Starting with pioneering work of J.L. Taylor, the theory was further developed by D.-V. Voiculescu, and established itself in recent years as a new and extremely active research area. The goal of the present paper is to establish a noncommutative analog of the Frobenius integrability theorem: we give necessary and sufficient conditions for higher order free noncommutative functions to have an antiderivative.
