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Georgios Stylogiannis

Publications and source records attributed to Georgios Stylogiannis.

14 recordsLinked to original sources
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

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

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↗
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

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↗
arXiv2021

Quasi-nilpotency of generalized Volterra operators on sequence spaces

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $\ell^p$ spaces $1<p<+\infty$ . Our main result is that when an analytic symbol $g$ is a multiplier for a weighted $\ell^p$ space, then the corresponding generalized Volterra operator $T_g$ is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $\{0\}.$ This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of $\ell^p$ spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $\ell^p$. We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on $\ell^p, 1<p<\infty$ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $\ell^2 $, extending a result of E. Ricard.

math.CA↗
arXiv2021

Hausdorff operators on Fock Spaces

Let $μ$ be a positive Borel measure on the positive real axis. We study the integral operator $$ \mathcal{H}_μ(f)(z)=\int_{0}^{\infty}\frac{1}{t}f\left(\frac{z}{t}\right)\,dμ(t),\quad z\in \mathbb{C}\,, $$ acting on the Fock spaces $F^{p}_α$, $p\in [1,\infty],\,α>0$. Its action is easily seen to be a coefficient multiplication by the moment sequence $$ μ_n= \int_{1}^{\infty}\frac{1}{t^{n+1}}\,dμ(t) . $$ We prove that \begin{equation*} ||\mathcal{H}_μ||_{F^{p}_α\to F^{p}_α}=\sup_{n\in\mathbb{N}}μ_n,\,\,\,\,\,1\leq p\leq \infty\,\,. \end{equation*} A little-o,condition describes the compactness of $\mathcal{H}_μ$ on every $F^{p}_α,\,p\in (1,\infty )$. In addition, we completely characterize the Schatten class membership of $\mathcal{H}_μ$.

math.FA↗