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<p>A representation of a right-angled Artin monoid is determined by a family of operators whose commutativity is dictated by a graph. 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 on right-angled Artin monoids, which generalizes Popescu’s notion of Cauchy and Poisson transforms for commuting families of row contractions. Finally, we prove that having <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-regular dilation is equivalent to the weak Brehmer’s condition plus the property (P), thereby establishing their equivalence to the generalized Brehmer’s condition.</p>

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Poisson transforms on right-angled Artin monoids

  • 期刊论文

  • 发表日期：

    2026年1月6日

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  • Boyu Li

摘要

A representation of a right-angled Artin monoid is determined by a family of operators whose commutativity is dictated by a graph. 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 on right-angled Artin monoids, which generalizes Popescu’s notion of Cauchy and Poisson transforms for commuting families of row contractions. Finally, we prove that having \(*\) ∗ -regular dilation is equivalent to the weak Brehmer’s condition plus the property (P), thereby establishing their equivalence to the generalized Brehmer’s condition.