Fixed point theorems for noncommutative functions

TL;DR

This paper develops fixed point theorems for noncommutative functions on matrices, extending classical results like Banach's theorem, and applies them to solve ODEs in noncommutative spaces.

Contribution

It introduces fixed point theorems for noncommutative matrix functions, including a new contractive mapping theorem, and applies these to noncommutative ODEs.

Findings

Established a fixed point theorem for matrix mappings respecting sizes and direct sums.

Proved a noncommutative version of the Banach Fixed Point Theorem.

Applied the results to ensure existence and uniqueness of solutions for noncommutative ODEs.

Abstract

We establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices. The conclusions are stronger if such a mapping also respects matrix similarities, i.e., is a noncommutative function. As a special case, we prove the corresponding contractive mapping theorem which can be viewed as a new version of the Banach Fixed Point Theorem. This result is then applied to prove the existence and uniqueness of a solution of the initial value problem for ODEs in noncommutative spaces. As a by-product of the ideas developed in this paper, we establish a noncommutative version of the principle of nested closed sets.