Cyclicity of stable matrix free polynomials over non-commutative operator unit balls

TL;DR

This paper characterizes cyclicity of stable non-commutative matrix polynomials over operator unit balls, establishing conditions for density of ideals and analyzing boundary behavior of NC rational functions.

Contribution

It introduces a stability criterion for cyclicity of matrix free polynomials and extends the theory to NC rational functions with boundary continuity.

Findings

Stable matrix free polynomials generate dense ideals if and only if they are non-singular everywhere.

Established a Neuwirth–Ginsberg–Newman inequality for stable NC polynomials.

Analyzed cyclicity of a new NC parallel sum function with boundary boundary properties.

Abstract

We consider the algebra of square matrices of bounded non-commutative (NC) functions over NC operator unit balls (unit balls corresponding to finite-dimensional operator spaces) and characterize cyclic matrix free polynomials with respect to the canonical weak-* topology. More precisely, we show that a matrix free polynomial generates a weak-* dense left/right ideal if and only if it is stable, i.e., non-singular at every point in the NC operator unit ball. To this end, we establish a version of the Neuwirth--Ginsberg--Newman inequality for stable matrix free polynomials. We combine our techniques with the theory of realizations to establish cyclicity of stable NC rational functions that are uniformly continuous across the boundary, and we recover known results about cyclicity of NC rational functions in the matrix-valued free Hardy space over the NC unit row-ball. Lastly, we introduce the NC parallel sum function: a stable NC rational function that is contractive over the NC bidisk, which cannot be extended uniformly across the boundary, and determine its cyclicity using properties of accretive operators.