Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914255382511616 |
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| author | Nailwal, Rajkamal Zalar, Aljaž |
| author_facet | Nailwal, Rajkamal Zalar, Aljaž |
| contents | For $α\in \mathbb{R},$ we consider the scale of function spaces, namely the Dirichlet-type space $\mathcal{D}_α$ consisting of holomorphic functions on the unit bidisk $\mathbb{D}^2$, $f(z,w)=\sum_{k,l=0}^{\infty}a_{kl}z^kw^l$ such that $$\sum_{k,l=0}^{\infty}(k+l+1)^α|a_{kl}|^2 < \infty.$$ In this paper, we solve an open problem posed in \cite[Open problem~1]{Z25}, which asks whether the polynomial $2 - z_1 - z_2$ is cyclic in $\mathcal{D}_α$ for $\frac{3}{2}<α\le 2$. We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $\mathcal{D}_α$. In addition, we establish several properties of cyclic functions and present alternative proofs of some cases previously obtained by P. T. Ziarati. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13441 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk Nailwal, Rajkamal Zalar, Aljaž Functional Analysis Primary 47A13, 32A37, Secondary 32A60 46E20 For $α\in \mathbb{R},$ we consider the scale of function spaces, namely the Dirichlet-type space $\mathcal{D}_α$ consisting of holomorphic functions on the unit bidisk $\mathbb{D}^2$, $f(z,w)=\sum_{k,l=0}^{\infty}a_{kl}z^kw^l$ such that $$\sum_{k,l=0}^{\infty}(k+l+1)^α|a_{kl}|^2 < \infty.$$ In this paper, we solve an open problem posed in \cite[Open problem~1]{Z25}, which asks whether the polynomial $2 - z_1 - z_2$ is cyclic in $\mathcal{D}_α$ for $\frac{3}{2}<α\le 2$. We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $\mathcal{D}_α$. In addition, we establish several properties of cyclic functions and present alternative proofs of some cases previously obtained by P. T. Ziarati. |
| title | Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk |
| topic | Functional Analysis Primary 47A13, 32A37, Secondary 32A60 46E20 |
| url | https://arxiv.org/abs/2511.13441 |