Divided Differences and Multivariate Holomorphic Calculus

TL;DR

This paper develops a simple, concrete extension of multivariate holomorphic functional calculus for tuples in Banach algebras, enabling easier derivation of properties and applications to noncommutative analysis.

Contribution

It introduces a naive but effective extension of multivariate holomorphic calculus that simplifies proofs and applications in noncommutative functional analysis.

Findings

Provides a straightforward proof of the Connes-Moscovici Rearrangement Lemma.

Shows the Daletski-Krein noncommutative Taylor expansion as a consequence of the calculus.

Derives Magnus' Theorem for nonlinear differential equations from the framework.

Abstract

We review the multivariate holomorphic functional calculus for tuples in a commutative Banach algebra and establish a simple "na\"ive" extension to commuting tuples in a general Banach algebra. The approach is na\"ive in the sense that the na\"ively defined joint spectrum maybe too big. The advantage of the approach is that the functional calculus then is given by a simple concrete formula from which all its continuity properties can easily be derived. We apply this framework to multivariate functions arising as divided differences of a univariate function. This provides a rich set of examples to which our na\"ive calculus applies. Foremost, we offer a natural and straightforward proof of the Connes-Moscovici Rearrangement Lemma in the context of the multivariate holomorphic functional calculus. Secondly, we show that the Daletski-Krein type noncommutative Taylor expansion is a natural consequence of our calculus. Also Magnus' Theorem which gives a nonlinear differential equation for the $\log$ of the solutions to a linear matrix ODE follows naturally and easily from our calculus. Finally, we collect various combinatorial related formulas.