Noncommutative formal power series and noncommutative functions

In various applications of formal power series, their evaluations on linear operators (acting on an infinite-dimensional Hilbert space) or on square matrices (of any size or of size large enough) play an important role and allow one to develop a noncommutative analog of analytic function theory. On the other hand, functions defined on square matrices of any size which respect direct sums and similarities and satisfy a local boundedness condition behave in many ways as analytic functions and have power series expansions - a noncommutative analogue of Taylor series. We will discuss convergence of noncommutative power series and analyticity of noncommutative functions.