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Cyclic polynomials in anisotropic Dirichlet spaces

  • Greg Knese
  • Łukasz KosińskiEmail author
  • Thomas J. Ransford
  • Alan A. Sola
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Abstract

Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions \(f(z_1,z_2):=\sum_{k,l\geq0}a_{kl}z_1^kz_2^l\) such that
$$\sum_{^{k,l\geq0}}(k+1)^{\alpha_1}(l+1)^{\alpha_2}|a_{kl}|^{2}<\infty.$$
Here the parameters α1, α2 are arbitrary real numbers. We characterize the polynomials that are cyclic for the shift operators on this space. More precisely, we show that, given an irreducible polynomial p(z1, z2) depending on both z1 and z2 and having no zeros in the bidisk:
  • if α1 + α2 ≤ 1, then p is cyclic;

  • if α1 + α2 > 1 and min{α1, α2} ≤ 1, then p is cyclic if and only if it has finitely many zeros in the two-torus \(\mathbb{T}^2\);

  • if min{α1, α2} > 1, then p is cyclic if and only if it has no zeros in \(\mathbb{T}^2\).

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Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Greg Knese
    • 1
  • Łukasz Kosiński
    • 2
    Email author
  • Thomas J. Ransford
    • 3
  • Alan A. Sola
    • 4
  1. 1.Department of MathematicsWashington University in St. LouisSt. LouisUSA
  2. 2.Institute of MathematicsJagiellonian UniversityKrakówPoland
  3. 3.Département de Mathématiques et de StatistiqueUniversité Laval, Pavillon Alexandre-VachonQuébec CityCanada
  4. 4.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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