Kaliuzhnyi-Verbovetskyi D. Foundations of the theory of free noncommutative functions and some it is applications in algebra and analysis

The foundations of the theory of functions of free noncommuting variables are systematically developed, in complete natural generality. This theory offers the unified way to view many free noncommutative objects that arise in different branches of mathematics. A noncommutative difference-differential calculus is constructed, up to the noncommutative Taylor formula. Necessary and sufficient conditions for the coefficients of the noncommutative power series are obtained to make the Taylor coefficients of the noncommutative function. It is proved that a noncommutative function that is a polynomial of matrix entries for each size of matrices, of bounded degree, must be a noncommutative polynomial. It is proved that a locally bounded noncommutative function must be analytic. The theory of convergence for noncommutative power series with matrix center is developed. The dissertation also solves several problems in special classes of noncommutative functions.