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Carlo Bellavita

Publications and source records attributed to Carlo Bellavita.

At least 19 recordsLinked to original sources
arXiv2026

Interchanging limits in the mutual information of two intervals separated by one site

In the paper J. Phys. A: Math. Theor. 53 (2020), 345303, the limiting mutual information between two intervals separated by one lattice site in a free-fermion chain was evaluated. The calculation replaces the entropy function by a regularized version depending on $\varepsilon>0$, takes the block lengths to infinity, and only then lets $\varepsilon$ tend to zero. We prove that the opposite order gives the same constant $2\log2-1$. Moreover, we show that the same limit is obtained whenever both block lengths tend to infinity and $\varepsilon$ tends to zero simultaneously, with no restriction on their relative rates. In fact, the error caused by the regularization is bounded by $(1+\varepsilon)\log(1+\varepsilon)-\varepsilon\log\varepsilon$, uniformly in both block lengths.

math.FA↗
arXiv2026

A discrete Mellin calculus in the Toeplitz algebra

We prove that the classical Cesàro operator belongs to the Toeplitz algebra, providing an independent solution to a question raised by Barría and Halmos. Our approach is based on a discrete Mellin calculus for the sampled-ratio matrices \[ W(κ)_{jk} = \frac{1}{j+1}\, κ\!\left(\frac{k+1}{j+1}\right). \] For a natural algebra of kernels $A$, we prove that this quantization is multiplicative modulo Hilbert--Schmidt operators, \[ W(κ)W(η) - W(κ\star η) \in S_2, \qquad κ,η\in A. \] We further show that every operator $W(κ)$, $κ\in A$, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel $κ= \mathbf 1_{(0,1]}$, this resolves the Barr\'ıa--Halmos question as a special case of the general framework.

math.FA↗
arXiv2026

Discrete random Clark measures and associated inner functions

We study a class of random inner functions $φ$ whose Clark measure at $1$ is the weighted sum of point masses supported on independent uniformly distributed points of $\mathbb T$. Our first result shows that $φ$ is almost surely a Blaschke product. We then investigate when $φ$ admits angular derivative almost surely and we provide a $0 - 1$ law. These conditions have a direct interpretation in terms of the other Clark measures associated with $φ$. Finally, we obtain quantitative estimates for the zeros of $φ$, proving that, in suitable regimes, their distribution satisfies summability conditions stronger than the classical Blaschke condition.

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↗
arXiv2025

Carleson Measures, Vanishing Mean Oscillation and Critical Points

Given a finite positive Borel measure $μ$ in the open unit disc of the complex plane, we construct a bounded outer function $E$ whose boundary values have vanishing mean oscillation such that $|E| μ$ is a vanishing Carleson measure. As an application it is shown that given any function in a Hardy space, there exists a bounded analytic function in the unit disc whose boundary values have vanishing mean oscillation, with the same critical points and multiplicities.

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

Fractional Volterra-type operator induced by radial weight acting on Hardy space

Given a radial doubling weight $μ$ on the unit disc $\mathbb{D}$ of the complex plane and its odd moments $μ_{2n+1}=\int_0^1 s^{2n+1}μ(s)\, ds$, we consider the fractional derivative $$ D^μ(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}}z^n, $$ of a function $ f(z)=\sum_{n=0}^{\infty}\widehat{f}(n)z^n$ analytic in $\mathbb{D}$. We also consider the fractional integral operator $I^μ(f)(z)=\sum_{n=0}^{\infty} μ_{2n+1}\widehat{f}(n)z^n$, and the fractional Volterra-type operator $$ V_{μ,g}(f)(z)= I^μ(f\cdot D^μ(g))(z),\quad f\in\mathcal{H}(\mathbb{D}), $$ for any fixed $g\in\mathcal{H}(\mathbb{D})$. We prove that $V_{μ,g}$ is bounded (compact) on a Hardy space $H^p$, $0<p<\infty$, if and only if $g$ belongs to $\text{BMOA}$ ($\text{VMOA}$). Moreover, if $\int_0^1 \frac{\left(\int_r^1 μ(s)\, ds\right)^p}{(1-r)^2}\,dr=+\infty$, we prove that $V_{μ,g}$ belongs to the Schatten class $S_p(H^2)$ if and only if $g=0$. On the other hand, if $\frac{\left(\int_r^1 μ(s)\, ds\right)^p}{(1-r)^2}$ is a radial doubling weight it is proved that $V_{μ,g} \in S_p(H^2)$ if and only if $g$ belongs to the Besov space $B_p$. En route, we obtain descriptions of $H^p$, $\text{BMOA}$, $\text{VMOA}$ and $B_p$ in terms of the fractional derivative $D^μ$.

math.CV↗
arXiv2025

On Toeplitz operators on $H^1(\mathbb{C}^+)$

In this paper we consider Toeplitz operators with anti-analytic symbols on $H^1(\mathbb{C}^+)$. It is well known that there are no bounded Toeplitz operators $T_{\overlineΘ}\colon H^1(\mathbb{C}^+) \to H^1(\mathbb{C}^+)$, where $Θ\in H^\infty(\mathbb{C}^+)$. We consider the subspace $H^1_Θ=\left\lbrace f \in H^1(\mathbb{C}^+)\colon \int_{\mathbb{R}}f \overlineΘ=0\right\rbrace$ and show that it is natural to study the boundedness of $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$. We provide several different conditions equivalent to such boundedness. We prove that when $Θ=e^{iτ(\cdot)}$, with $τ>0$ $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$ is bounded. Finally, we discuss a number of related open questions.

math.FA↗
arXiv2024

Meromorphic Optimal domain of Integral Operators

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 domains coincide. This investigation contributes to a more comprehensive understanding of the holomorphic optimal domain of $T_g$ in $H^p$.

math.CV↗
arXiv2024

On the Hilbert matrix operator: a brief survey

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.

math.FA↗
arXiv2024

The spectral analysis of the difference quotient operator on model spaces

We conduct a spectral analysis of the difference quotient operator $Q^u_ζ$, associated with a boundary point $ζ\in \partial \mathbb{D}$, 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 offers a new characterization of the one-component property for inner function.

math.FA↗
arXiv2024

Hilbert matrix operator acting between conformally invariant spaces

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 $μ$ such that $\mathcal H$ is bounded from $H^\infty$ into the conformally invariant Dirichlet space $M(\mathcal{D}_μ)$. For particular measures $μ$, we also provide the norm of $\mathcal{H}$ from $H^\infty$ into $M(\mathcal{D}_μ)$.

math.FA↗
arXiv2024

Generalized Hilbert matrix operators acting on Bergman spaces

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

math.CV↗
arXiv2024

Optimal Domain of generalized Volterra operators

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

math.CV↗
arXiv2024

Generalized Hilbert operators arising from Hausdorff matrices

For a finite, positive, Borel measure $μ$ on $(0,1)$ we consider an infinite matrix $Γ_μ$, related to the classical Hausdorff matrix defined by the same measure $μ$, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When $μ$ is the Lebesgue measure, $Γ_μ$ reduces to the classical Hilbert matrix. We prove that the matrices $Γ_μ$ are not Hankel, unless $μ$ is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces $H^p, \, 1 \leq p < \infty$, and we study their compactness and complete continuity properties. In the case $2\leq p<\infty$, we are able to compute the exact value of the norm of the operator.

math.FA↗
arXiv2023

Embedding model and de Branges-Rovnyak spaces in Dirichlet spaces

In this paper we study embeddings between de Branges-Rovnyak spaces $H(b)$ and harmonically weighted Dirichlet spaces $\mathcal{D}(μ)$ 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 $\mathcal{D}_ζ$, and we discuss some applications.

math.FA↗