Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra

TL;DR

This paper explores Schur analysis within the Grassmann algebra completed under the 1-norm, introducing notions of positivity, and extending classical results like the Schur algorithm to this new algebraic setting.

Contribution

It develops a framework for Schur analysis in the Grassmann algebra, including invertibility, positivity, and extensions of classical concepts like Blaschke factors.

Findings

Established invertibility criteria in the completed Grassmann algebra

Defined a notion of positivity in this algebraic context

Extended classical Schur analysis results to the Grassmann algebra setting

Abstract

We begin a study of Schur analysis in the setting of the Grassmann algebra, when the latter is completed with respect to the $1$-norm. We focus on the rational case. We start with a theorem on invertibility in the completed algebra, and define a notion of positivity in this setting. We present a series of applications pertaining to Schur analysis, including a counterpart of the Schur algorithm, extension of matrices and rational functions. Other topics considered include Wiener algebra, reproducing kernels Banach modules, and Blaschke factors.