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Athanasios Beslikas

Publications and source records attributed to Athanasios Beslikas.

8 recordsLinked to original sources
arXiv2026

Critical cyclicity in Dirichlet-type spaces on the bidisk

Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}_{\beta}(\mathbb{D}^2)=\Bigg\{f(z_1,z_2)=\sum_{k,l}a_{kl}z_1^kz_2^l\in\mathcal{O}(\mathbb{D}^2): \sum_{k,l\ge 0}|a_{kl}|^2(k+l+1)^{\beta}<+\infty\Bigg\}.$$ Given $\beta_c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space $\mathcal D_{2}(\mathbb{D}^2)$ of the bidisk and is cyclic in $\mathcal D_\beta(\mathbb{D}^2)$ if and only if $\beta\leq \beta_{c}.$ We also show that the critical index satisfies $\beta_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}^2),$ where $\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}^2.$

math.CV↗
arXiv2026

Composition operators and Rational Inner Functions on the bidisc: A geometric approach

We study composition operators acting on the weighted Bergman spaces on the bidisc, i.e. $C_{\Phi}:A^2_{\beta}(\mathbb{D}^2)\to A^2_{\beta}(\mathbb{D}^2)$ where $\Phi$ is induced by rational inner functions (RIFs) or a RIF and a smooth function (mixed case). Our approach is geometric. Our main result is a uniform criterion for all $\beta\in(-1,0]$ that can be summarized as follows: Boundedness of the composition operator is equivalent to transversal intersection of the level sets for non-smooth symbols, under the assumption that if any tangential intersection occurs on the singularity it must be of high order. This extends the characterization of Bayart-Kosi\'nski to the non-smooth self maps of the bidisc. To reach our conclusions, we utilize results obtained by Anderson, Bergqvist, Bickel, Cima and Sola on Clark measures associated to RIFs and Puiseux factorizations.

math.CV↗
arXiv2025

On the membership of two-variable Rational Inner Functions in spaces of Dirichlet-type

We study membership of rational inner functions on the bidisk $\mathbb{D}^2$ in a scale of Dirichlet spaces considered by Bera, Chavan, and Ghara, and in higher-order variants of these spaces. We give a characterization for membership in terms of the geometric concept of contact order of a rational inner function at its singular points, and we further record some consequences and variants of our main result.

math.CV↗
arXiv2024

Composition operators and Rational Inner Functions on the bidisc

In the present article, composition operators induced by Rational Inner Functions on the bidisc $\mathbb{D}^2$ are studied, acting on the weighted Bergman space $A^2_{\beta}(\mathbb{D}^2).$ We prove that under mild conditions that Rational Inner Functions with one singularity on $\mathbb{T}^2$ induce unbounded composition operator on $A^2(\mathbb{D}^2).$ We also prove that under the condition of stability of the polynomial inducing the Rational Inner Function, the composition operator is bounded between two different Bergman spaces.

math.CV↗
arXiv2024

A note on composition operators on the bidisc

In this note we give a new sufficient condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterise the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak{D}_{\vec{a}}(\mathbb{D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb{D}^2$ of the form $\Phi(z_1,z_2)=(\phi_1(z_1),\phi_2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\Phi}:\mathfrak{D}(\mathbb{D}^2)\to A^2(\mathbb{D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the the Dirichlet space of the bidisc.

math.CV↗
arXiv2024

A class of symbols that induce bounded composition operators for Dirichlet-type spaces on the disc

In this note we study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi$ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

math.CV↗
arXiv2023

New Characterizations of $N(p,q,s)$-type spaces on the unit ball of $\mathbb{C}^n$

We provide some new characterizations of the N(p,q,s)-type spaces on the unit ball. Specifically, we give some Holland-Walsh-type characterizations for functions on the N(p,q,s) spaces, for specific values of the parameter $p$, that have not been found in the literature. The techniques and tools used throught this note are pretty standard.

math.CV↗