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Poisson transforms on right-angled Artin monoids

Boyu Li

TL;DR

This work extends Popescu's Cauchy and Poisson transforms to contractive representations of right-angled Artin monoids by introducing the weak Brehmer condition and property (P). It proves the Cauchy transform $C_{r,T}$ is bounded for $0\le r<1$, enabling the Poisson transform $\Phi_T$ and establishing a $*$-regular dilation via Stinespring dilation for representations satisfying these conditions. A key advance is reducing positivity checks to the connected components of the complement graph, with a bound depending on the clique number $\omega(\Gamma)$. In the complete multipartite case, the results recover Popescu's transform and open a path for multivariate operator theory on RA monoids.

Abstract

We introduce the notion of the weak Brehmer's condition and prove that the Cauchy transform for a representation of a right-angled Artin monoid is bounded under such conditions. As a result, we obtain the Poisson transform and $*$-regular dilation for a family of operators that satisfies the weak Brehmer's condition and the property (P). This generalizes Popescu's notion of Cauchy and Poisson transforms for commuting families of row contractions.

Poisson transforms on right-angled Artin monoids

TL;DR

This work extends Popescu's Cauchy and Poisson transforms to contractive representations of right-angled Artin monoids by introducing the weak Brehmer condition and property (P). It proves the Cauchy transform is bounded for , enabling the Poisson transform and establishing a -regular dilation via Stinespring dilation for representations satisfying these conditions. A key advance is reducing positivity checks to the connected components of the complement graph, with a bound depending on the clique number . In the complete multipartite case, the results recover Popescu's transform and open a path for multivariate operator theory on RA monoids.

Abstract

We introduce the notion of the weak Brehmer's condition and prove that the Cauchy transform for a representation of a right-angled Artin monoid is bounded under such conditions. As a result, we obtain the Poisson transform and -regular dilation for a family of operators that satisfies the weak Brehmer's condition and the property (P). This generalizes Popescu's notion of Cauchy and Poisson transforms for commuting families of row contractions.

Paper Structure

This paper contains 5 sections, 13 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Right-angled Artin monoids and right LCM monoids
  3. Brehmer conditions and regular dilation
  4. Cauchy transform
  5. Poisson transform

Key Result

Lemma 2.2

Let $F\subset A_\Gamma^+$ be a finite subset and suppose $\vee F\neq\infty$. Suppose $I$ is an initial vertex for some $x\in F$. Then for each $y\in F$, either $I$ is an initial vertex of $y$ or $I$ is adjacent to every vertex of syllables of $y$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (32)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Definition 3.3
  • Example 3.4
  • Example 3.5
  • Lemma 4.1
  • proof
  • ...and 22 more