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PreprintCC BY 4.0arXiv

Critical cyclicity in Dirichlet-type spaces on the bidisk

  • Athanasios Beslikas
  • Pouriya Torkinejad Ziarati
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Abstract

Consider the Dirichlet-type spaces on the bidisc defined by π’ŸΞ²(𝔻2)={f(z1,z2)=βˆ‘k,laklz1kz2l∈π’ͺ(𝔻2): βˆ‘k,lβ‰₯ 0|akl|2(k+l+1)Ξ²<+∞}. Given Ξ²c∈(0,2], we construct a function f that belongs to the Dirichlet-type space π’Ÿ2(𝔻2) of the bidisk and is cyclic in π’ŸΞ²(𝔻2) if and only if β≀ Ξ²c. We also show that the critical index satisfies Ξ²c=2βˆ’dimH(𝒡(f)∩ 𝕋2), where dimH(𝒡(f)∩ 𝕋2) is the Hausdorff dimension of the zero set of the function f on the two-torus 𝕋2.

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Cite

BibTeXRIS
BibTeX
@misc{beslikas2026critical,
  title = {{Critical cyclicity in Dirichlet-type spaces on the bidisk}},
  author = {Athanasios Beslikas and Pouriya Torkinejad Ziarati},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09952},
  eprint = {2609.09952},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09952},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Critical cyclicity in Dirichlet-type spaces on the bidisk
AU  - Athanasios Beslikas
AU  - Pouriya Torkinejad Ziarati
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.09952
UR  - https://arxiv.org/abs/2609.09952
AB  - Consider the Dirichlet-type spaces on the bidisc defined by π’Ÿ_Ξ²(𝔻^2)={f(z_1,z_2)=βˆ‘_k,la_klz_1^kz_2^l∈π’ͺ(𝔻^2): βˆ‘_k,lβ‰₯ 0|a_kl|^2(k+l+1)^Ξ²<+∞}. Given Ξ²_c∈(0,2], we construct a function f that belongs to the Dirichlet-type space π’Ÿ_2(𝔻^2) of the bidisk and is cyclic in π’Ÿ_Ξ²(𝔻^2) if and only if β≀ Ξ²_c. We also show that the critical index satisfies Ξ²_c=2βˆ’dim_H(𝒡(f)∩ 𝕋^2), where dim_H(𝒡(f)∩ 𝕋^2) is the Hausdorff dimension of the zero set of the function f on the two-torus 𝕋^2.
ER  -