arXiv subjects

Carlos Gómez-Cabello

Publications and source records attributed to Carlos Gómez-Cabello.

6 recordsLinked to original sources

arXiv2026

On frequencies of parabolic Koenigs domains

Let $(φ_t)_{t\geq 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $Ω= h(\mathbb{D})$. We study the point spectrum $σ_p(Δ\mid_{H^p})$ of $Δ$, the infinitesimal generator of the $C_0$-semigroup $(C_{φ_t})_{t\geq 0}$ of composition operators on $H^p$. This reduces to characterizing the frequencies of $Ω$. That is, those $λ\in \mathbb{C}$ such that $e^{λh} \in H^p$. We derive containment relations for $σ_p(Δ\mid_{H^p})$ and provide sufficient conditions for its complete characterization. Our approach relies heavily on the geometric properties of $Ω$ and on careful estimates of the harmonic measure of some boundary subsets of $Ω$. We conclude with some consequences regarding the spectrum of the composition operators $(C_{φ_t})_{t\geq 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups.

math.CV↗

arXiv2024

Volterra operator acting on Bergman spaces of Dirichlet series

Since their introduction in 1997, the Hardy spaces of Dirichlet series have been broadly and deeply studied. The increasing interest sparked by these Banach spaces of Dirichlet series motivated the introduction of new such spaces, as the Bergman spaces of Dirichlet series $\mathcal{A}^p_μ$ here considered, where $μ$ is a probability measure on $(0,\infty)$. Similarly, recent lines of research have focused their attention on the study of some classical operators acting on these spaces, as it is the case of the Volterra operator $T_g$. In this work, we introduce a new family of Bloch spaces of Dirichlet series, the $\text{Bloch}_μ$-spaces, and study some of its most essential properties. Using these spaces we are able to provide a sufficient condition for the Volterra operator $T_g$ to act boundedly on the Bergman spaces $\mathcal{A}^p_μ$. We also establish a necessary condition for a specific choice of the probability measures $μ$. Sufficient and necessary conditions for compactness are also proven. The membership in Schatten classes is studied as well. Eventually, a radicality result is established for Bloch spaces of Dirichlet series.

math.FA↗

arXiv2024

On the uniform convergence of continuous semigroups

Let $Ω$ be a region in the complex plane $\mathbb C$ and let $\{Φ_t \}_{t\ge 0}$ be a continuous semigroup of functions on $Ω$; that is, $Φ_t:Ω\toΩ$ is holomorphic for every $t\ge 0$, $Φ_0(z)=z$, for every $z\inΩ$, $Φ_t\circΦ_s=Φ_{s+t}$, for every $s$, $t\ge 0$, and \[ Φ_t(z)\to z\,,\quad t\to0^+, \] uniformly on compact subsets of $Ω$. Despite this definition only requires the uniform convergence on compact subsets, P. Gumenyuk proved in 2014 that, when $Ω$ is the unit disc, the convergence is uniform on the whole $\mathbb D$. In this paper, we enhance Gumenyuk's result by proving that for every continuous semigroup $\{Φ_t\}_{t\ge 0}$ on $\mathbb D$ we have $$ \sup_{z\in\mathbb D}|Φ_t(z)-z|= O(\sqrt t), \ t\to0^+. $$ In addition, we provide an example showing that $O(\sqrt t)$ is the best possible rate of uniform convergence valid for all semigroups on $\mathbb D$. When $Ω$ is the right half-plane $\mathbb C_+$, we consider semigroups $\{Φ_t\}$ with $\infty$ as its Denjoy-Wolff point. It is not difficult to show that Gumenyuk's result is no longer true for these semigroups. Our second result characterises when such continuous semigroups converge uniformly to the identity, as $t\to0^+$, in terms of their infinitesimal generators. Namely, this convergence holds if and only if the infinitesimal generator of the semigroup is bounded in the half-plane $\{z\in \mathbb C:\, \Re z>1\}$. In this case, we can also prove that the rate of convergence is again $O(\sqrt{t})$, as $t\to0^+$. An example of application of this result is when the semigroup is in the Gordon-Hedenmalm class (the one producing bounded composition operators on Hardy spaces of Dirichlet series). An important ingredient in the proofs of these results is harmonic measure, which we have done through a classic result of M. Lavrentiev.

math.CV↗

arXiv2024

Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series

The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit disc. In this sense, Bergman spaces were introduced several years ago contributing to broaden the picture of this theory. In the paper presenting these new family of spaces, some of its most essential questions were considered. Among them, some partial estimates of the norm of the pointwise evaluation functional were given. In this work, we introduce a version of the Riemann-Liouville semigroup acting on these spaces, and, with this new tool, we are able to estimate the norm of this functional.

math.CV↗

arXiv2023

Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series $\mathcal{A}(\mathcal{C}_{+})$ consists on those Dirichlet series convergent in the right half-plane $\mathcal{C}_{+}$ and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols $Φ:\mathcal{C}_{+}\to\mathcal{C}_{+}$ giving rise to bounded composition operators $\mathit{C}_Φ$ in $\mathcal{A}(\mathcal{C}_{+})$ and denote this class by $\mathcal{G}_{\mathcal{A}}$. We also characterise when the operator $\mathit{C}_Φ$ is compact in $\mathcal{A}(\mathit{C}_{+})$. As a byproduct, we show that the weak compactness is equivalent to the compactness for $\mathit{C}_Φ$. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions $\{Φ_{t}\}$ in the class $\mathcal{G}_{\mathcal{A}}$ and strongly continuous semigroups of composition operators $\{T_{t}\}$, $T_{t}f=f\circΦ_{t}$, $f\in\mathcal{A}(\mathcal{C}_{+})$. We conclude providing examples showing the differences between the symbols of bounded composition operators in $\mathcal{A}(\mathcal{C}_{+})$ and the Hardy spaces of Dirichlet series $\mathcal{H}^{p}$ and $\mathcal{H}^{\infty}$.

math.FA↗

arXiv2022

Semigroups of composition operators on Hardy spaces of Dirichlet series

We consider continuous semigroups of analytic functions $\{Φ_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $Φ:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We show that there is a one-to-one correspondence between continuous semigroups $\{Φ_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circΦ_t$, $f\in\mathcal{H}^2$. We extend these results for the range $p\in[1,\infty)$. For the case $p=\infty$, we prove that there is no non-trivial strongly continuous semigroup of composition operators in $\mathcal{H}^\infty$. We characterize the infinitesimal generators of continuous semigroups in the class $\mathcal G$ as those Dirichlet series sending $\mathbb C_{+}$ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.

math.FA↗