arXiv subjects

Eugenio Dellepiane

Publications and source records attributed to Eugenio Dellepiane.

7 recordsLinked to original sources

arXiv2026

Hardy spaces of discrete holomorphic functions on the upper half-lattice

We develop a theory of Hardy spaces $H^p$ of discrete holomorphic functions on the upper half-lattice, within the classical framework of discrete holomorphicity on the square lattice. We prove Cauchy and Poisson reproducing formulas, establish a boundary norm identity, and obtain Paley--Wiener type characterizations for these spaces. In the Hilbert space case, we describe the associated reproducing kernel and Szegő projection, and we compare the discrete theory with the classical Hardy space on the upper half-plane through a family of discrete holomorphic approximants of classical $H^2$-functions. We also prove duality results for $H^p$, $1<p<\infty$, establish uniqueness and sampling results on horizontal lines, and introduce Bergman-type spaces, comparing two natural weighted scales.

math.CV↗

arXiv2026

Rhaly operators on weighted Hardy spaces and factorable matrices

In this paper, we study properties of Rhaly operators acting on weighted Hardy spaces $H^2(ω)$. This problem is intimately related to the study of factorable matrices on $\ell^2$. In particular, our main contributions are in the study of Schatten class properties of such operators.

math.FA↗

arXiv2026

Orthogonal polynomials in de Branges--Rovnyak spaces

Given a function $b$, holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges--Rovnyak space $H(b)$. We explore representations of such spaces via descriptions of the corresponding families of orthogonal polynomials. We find relevant structures in the linear systems involved in a diversity of cases when $b$ is rational. We also establish a form of invariance under some composition operators on $H(b)$ spaces.

math.CV↗

arXiv2025

Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces

Harmonically weighted Dirichlet spaces $\mathcal{D}_μ$ and de Branges_Rovnyak spaces $\mathcal{H}(b)$ are two fundamental structures in analytic function theory exhibiting rich and often complementary properties. The question of when these spaces coincide, first raised and solved in Sarason's groundbreaking work in 1997 when $μ$ is a single Dirac mass, is thus of fundamental importance in operator theory and analytic function spaces. In this paper, we focus on spaces $\mathcal{H}(b)$ with symbol $b = (1+u)/2$, where $u$ is a one-component inner function. While previous results extended Sarason's work to finitely supported measures $μ$, the symbols we consider here give a natural framework to go beyond finiteness of the support. In our setting, we provide a complete characterization of measures $μ$ for which $\mathcal{H}(b) = \mathcal{D}_μ$, thereby resolving the long-standing open problem of constructing harmonically weighted Dirichlet spaces $\mathcal{D}_μ$ associated with measures $μ$ of infinite support that are also $\mathcal{H}(b)$ spaces. As a central ingredient to prove this result and which is of independent interest, we establish a $T(1)$-type result for the Cauchy transform on $L^2(σ)$, where $σ$ denotes the Clark measure associated with a one-component inner function $u$. Another notable result is a perturbation theorem for one-component inner functions that allows us to present a large class of function spaces satisfying $\mathcal{H}(b)=\mathcal{D}_μ$. Furthermore, we settle the Brown--Shields conjecture within this setting.

math.CV↗

arXiv2025

Boundedness, compactness and Schatten class for Rhaly matrices

In this article we present new proofs for the boundedness and the compactness on $\ell^2$ of the Rhaly matrices, also known as terraced matrices. We completely characterize when such matrices belong to the Schatten class $\mathcal{S}^q(\ell^2)$, for $1<q<\infty$. Finally, we apply our results to study the Hadamard multipliers in weighted Dirichlet spaces, answering a question left open by Mashreghi-Ransford.

math.FA↗

arXiv2025

On the equality of De Branges-Rovnyak and Dirichlet spaces

This work is devoted to the comparison of de Branges--Rovnyak $H(b)$ spaces harmonically weighted Dirichlet spaces $\mathcal{D}_μ$. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces $\mathcal{D}_μ$, when $μ$ is a finite sum of atoms. This is a generalization of a previous result by Costara--Ransford \cite{costara2013}: we make no assumptions on the Pythagorean pair $(b,a)$, and we produce new examples.

math.FA↗

arXiv2024

The Cesàro operator on local Dirichlet spaces

The family of Cesàro operators $σ_n^α$, $n \geq 0$ and $α\in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_ζ$. It is well-established that they provide a linear approximation scheme when $α> \frac{1}{2}$, with the threshold value $α= \frac{1}{2}$ being optimal. We strengthen this result by deriving precise asymptotic values for the norm of these operators when $α\leq \frac{1}{2}$, corresponding to the breakdown of approximation schemes. Additionally, we establish upper and lower estimates for the norm when $α> \frac{1}{2}$.

math.FA↗