Carlo Bellavita

University of Barcelona | UB · Department of Applied Mathematics and Analysis

For $g\in BMOA$, we introduce the meromorphic optimal domain $(T_g,H^p)$, i.e. the space containing the meromorphic functions that are mapped under the action of the generalized Volterra operator $T_g$ into the Hardy space $H^p$. We investigate its properties and characterize for which $g_1,g_2 \in BMOA$ the corresponding meromorphic optimal domain...

This article aims to explore the most recent developments in the study of the Hilbert matrix, acting as an operator on spaces of analytic functions and sequence spaces. We present the latest advances in this area, aiming to provide a concise overview for researchers interested in delving into the captivating theory of operator matrices.

We conduct a spectral analysis of the difference quotient operator $Q^u_\zeta$, associated with a boundary point $\zeta \in \partial \mathbb{D}$, on the model space $K_u$. We describe the operator's spectrum and provide both upper and lower estimates for its norm, and furthermore discussing the sharpness of these bounds. Notably, the upper estimate...

In this article we consider the generalized integral operators acting on the Hilbert space $H^2$. We characterize when these operators are uniform, strong and weakly asymptotic Toeplitz and Hankel operators. Moreover we completely describe the symbols $g$ for which these operators are essentially Hankel and essentially Toeplitz.

For a finite, positive Borel measure μ \mu on ( 0 , 1 ) (0,1) we consider an infinite matrix Γ μ \Gamma _\mu , related to the classical Hausdorff matrix defined by the same measure μ \mu , in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When μ \mu is the Lebesgue measure, Γ μ \Gamma _\mu reduces to the classical H...

In this article we study the action of the the Hilbert matrix operator $\mathcal H$ from the space of bounded analytic functions into conformally invariant Banach spaces. In particular, we describe the norm of $\mathcal{H}$ from $H^\infty$ into $\text{BMOA}$ and we characterize the positive Borel measures $\mu$ such that $\mathcal H$ is bounded fro...

In this article we study the generalized Hilbert matrix operator $\Gamma_\mu$ acting on the Bergman spaces $A^p$ of the unit disc for $1\leq p<\infty$. In particular, we characterize the measures $\mu$ for which the operator $\Gamma_\mu$ is bounded and we provide estimates of its operator norm. Finally, we also describe when $\Gamma_\mu$ is compact...

For g in BMOA, we consider the generalized Volterra operator Tg acting on Hardy spaces H p. This article aims to study the largest space of analytic functions, which is mapped by Tg into the Hardy space H p. We call this space the optimal domain of Tg and we describe its structural properties. Motivation for this comes from the work of G. Curbera a...

In this paper we study embeddings between de Branges-Rovnyak spaces H(b) and harmonically weighted Dirichlet spaces D(µ) in terms of the boundary spectrum of b and the support of the measure µ, by using elementary reproducing kernel estimates. We completely characterize the embedding between the model spaces K u and the local Dirichlet spaces D ζ ,...

In this paper we deal with the problem of describing the dual space $(B^1_\kappa)^*$ of the Bernstein space $B^1_\kappa$, that is the space of entire functions of exponential type at most $\kappa>0$ whose restriction to the real line is Lebesgue integrable. We provide several characterisations, showing that such dual space can be described as a quo...

For a finite positive Borel measure $\mu$ on $(0,1)$ we consider an infinite matrix $\Gamma_\mu$ which is connected to the classical Hausdorff matrix, corresponding to the same measure $\mu$, in the same algebraic way that the Ces\'aro matrix is related to the Hilbert matrix. We prove that the matrices $\Gamma_\mu$ are not Hankel, unless $\mu$ is a...

Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the corresponding de Branges–Rovnyak spaces and the last one concerns the associated Aleksandrov–Clark measure.

In the previous work Bellavita (Complex Anal. Oper. Theory 15: 96, 2021) we found some necessary conditions for the boundedness of the translation operator \(T_\zeta\) in the de Branges space \({{\mathcal {H}}}(E)\). In that case we made use of the Carleson measures for the associated model space. In this work we start from the Pancherel-Polya ineq...

Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the corresponding de Branges-Rovnyak spaces and the last one concerns the associated Clark measure.

The translation operator is bounded in the Paley–Wiener spaces and, more generally, in the Bernstein spaces. The goal of this paper is to find some necessary conditions for the boundedness of the translation operator in the de Branges spaces, of which the Paley–Wiener spaces are special cases. Indeed, if the vertical translation operator Tτ defined...

In this paper we study the continuity of the embedding operator ℓ : ℋp(E) ↪ ℋ q(E) when 0 < p < q ⩽ ∞. The necessary and sufficient condition has already been described in [10] if p > 1. In this work, we address the problem when p = 1, using a new approach, but asking some additional hypothesis about the Hermite-Biehler function E. We give also a d...

In this paper we study the continuity of the embedding operator between p-de Branges spaces. We describe some properties the Hermite-Biehler function E has to satisfy so that H^p(E) -> H^q(E) is continuous if p <= q. Considering how we prove that the embedding operator is continuous in the Bernstein spaces, we gain some sufficient and some necessar...