Inverse and Implicit Function Theorems for Noncommutative Functions on Operator Domains

TL;DR

This paper develops inverse and implicit function theorems for noncommutative functions on operator domains, extending classical results to infinite-dimensional operator settings with new dilation-theoretic techniques.

Contribution

It introduces a framework for noncommutative functions on operator domains and establishes inverse and implicit function theorems in this context, relaxing derivative conditions.

Findings

Established inverse and implicit function theorems for noncommutative functions.

Used dilation theory to relax derivative assumptions from boundedness below to injectivity.

Extended classical analysis results to infinite-dimensional operator settings.

Abstract

We introduce a notion of a noncommutative function defined on a domain of $d$-tuples of bounded operators on an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to merely injectivity.