Integrability of Free Noncommutative Functions

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.