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

Jeet Sampat, Maximilian Tornes

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.

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

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.
Paper Structure (18 sections, 21 theorems, 121 equations)

This paper contains 18 sections, 21 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Commutative cyclic polynomials
  3. Free analysis
  4. Main results
  5. Preliminaries
  6. Spectral radius corresponding to an operator space
  7. Stable associativity and linearization
  8. A free Neuwirth--Ginsberg--Newman-type inequality
  9. Further analysis of the bounds
  10. Cyclicity of matrix free polynomials
  11. Canonical weak-* topology
  12. Weak-* cyclicity
  13. Cyclic polynomials in the free Hardy space
  14. Cyclicity of NC rational functions
  15. NC parallel sum function
  16. ...and 3 more sections

Key Result

Theorem A

If $P \in M_k(\mathbb{C} \langle Z \rangle)$ is a $Q$-stable atom, then

Theorems & Definitions (49)

  • Example 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Theorem 2.2: Corollary 2.12, Shalit-Shamovich-spec-rad
  • Theorem 2.3: Theorem 3.4, Shalit-Shamovich-spec-rad
  • Definition 2.4
  • Lemma 2.5: Lemma 4.2, Helton-Klep-Volcic-Free-factor
  • Example 2.6
  • ...and 39 more