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About 3,329 papers for “analytic function spaces”

Multiplicative operators on analytic function spaces

Kanha Behera, Jun‐Ming Liu, Perumal Muthukumar · ArXiv.org · 2025

H. J. Schwartz proved in his thesis (1969) that a nonzero bounded operator on Hardy spaces $(H^p, 1\leq p\leq\infty)$ is almost multiplicative if and only if it is a composition operator. But, his proof has a gap. In this article, we show that his result is not correct for $H^\infty$ and we fill the gap for $H^p, 1\leq p<\infty.$ Further, we prove that on several classical spaces such as the Bloch space, the little Bloch space, Besov spaces $B_p$ for $p>1$, and weighted Bergman spaces an operator is almost multiplicative if and only if it is a composition operator. Finally, we give a complete characterization of those composition operators that are multiplicative with respect to the Duhamel product of analytic functions.

Cited by 0Open access

Coupled Maps and Analytic Function Spaces

Hans Henrik Rugh · 2001

ABSTRACT. – We consider ergodic properties of weakly coupled analytic and expanding circle maps. For weak enough coupling a natural ergodic measure exists and exhibits exponent decay of time correlations. The marginal densities of the natural measure are analytic. A spatial decay of correlations (e.g. polynomial) in these densities may arise from a similar spatial decay of the couplings. The space of couplings and observables is a Banach algebra of analytic functions of infinitely many variables. This algebra acts upon a Banach module of complex measures with analytic marginal densities. A Perron–Frobenius type operator acts on the Banach module and we apply a re-summation technique to derive uniform bounds for this operator. Explicit bounds are calculated for some examples. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Nous étudions des propriétés ergodiques des applications analytiques et dilatantes du cercle, faiblement couplées. Lorsque le couplage est assez petit, il y a une mesure naturelle ergodique pour laquelle les corrélations en temps décroissent exponentiellement vite. Les densités marginales de la mesure naturelle sont analytiques. Une décroissance spatiale des corrélations (par exemple polynomiale) de ces densités peut être la manifestation d’une décroissance spatiale similaire pour le couplage. L’espace de couplages et d’observables est une algèbre de Banach de fonctions analytiques d’un nombre infini de variables. Cette algèbre agit sur un module de Banach de mesures complexes qui possèdent des densités marginales analytiques. Un opérateur de type Perron–Frobenius agit sur le module de Banach et nous appliquerons une technique de resommation pour en déduire des bornes uniformes pour cet opérateur. Nous calculons des bornes explicites pour quelques exemples. 2002 Éditions scientifiques et médicales Elsevier SAS

Cited by 3

Mixed norms and analytic function spaces

Stephen M. Buckley · MURAL - Maynooth University Research Archive Library (National University of Ireland, Maynooth) · 2000

We de�ne and investigate general mixed-norm type sequence spaces, and strengthen inequalities of Hardy-Littlewood, Hausdor�-Young, and others.

Cited by 8

MIXED NORMS AND ANALYTIC FUNCTION SPACES

M. Buckley · 2009

We define and investigate general mixed-norm type sequence spaces, and strengthen inequalities of Hardy-Littlewood, Hausdorff-Young, and others. Since the time of Hardy and Littlewood, mixed-norm and related spaces have been used to study Taylor coefficients in function spaces on the unit disk, and later to study multipliers between such spaces; for example, see [9], [2], [13], [3], [5], [6], and the many references in those papers. Here, we define very general

Cited by 0

Generalized Hilbert operator on analytic function spaces

Simi Bhuyan, Sunanda Naik · Arab Journal of Mathematical Sciences · 2025

Purpose The main objective of this work is to study the boundedness property of a generalized Hilbert operator Hβ for β ≥ 0 defined by, if f(z)=∑n=0∞anzn be any analytic function on the unit disk D then Hβf(z)=∑n=0∞∑k=0∞Γ(n+β+1)Γ(n+k+1)Γ(n+1)Γ(n+k+β+2)akzn on few analytic function spaces such as Hardy space H1, weighted Hardy spaces HαP and the Bloch space. Design/methodology/approach The authors shall use a few well-known results obtained by using Hardy’s inequality and the Hardy–Littlewood theorem to find the coefficient conditions of an analytic function f on the unit disk D so that f is in Hardy spaces. Also, the authors use many properties of the Gaussian hypergeometric function to prove the main results. Findings We find conditions on the parameter β and establish criteria for the boundedness of Hβ on certain function spaces such as Hardy space H1, weighted Hardy spaces HαP and the Bloch space. Originality/value In particular, β = 0 gives the classical Hilbert operator H. So our results extend some results of Lanucha et al. [10]. The authors use of the Gaussian hypergeometric function in the proof is a different approach.

Cited by 0Open access

Operator Theory on Univariate Analytic Function Spaces

Yan, Yi · KU ScholarWorks (The University of Kansas) · 2021

The goal of this thesis is to study certain aspects of Toeplitz, Hankel, and composition operators, as well as the associated operator algebras, on various Hardy and Bergman spaces of univariate analytic functions. The work belongs to the general area of function theoretic operator theory, with added flavors of uniform function algebras and C*-algebras. It is hoped that the results obtained reveal new and perhaps interesting connections between properties of the symbol functions and those of the induced operators. It is also hoped that some of the methods and techniques developed in this research could help solve other problems in the theory of Toeplitz and Hankel operators beyond the scope of this thesis.

Cited by 0

Nonlinear differential equation and analytic function spaces

Hao Li, Songxiao Li · Complex Variables and Elliptic Equations · 2017

In this paper we consider the nonlinear complex differential equationwhere , , are analytic in the unit disk , for all . We investigate this nonlinear differential equation from two aspects. On the one hand, we provide some sufficient conditions on coefficients such that the analytic solutions of this equation belong to the space. On the other hand, we find growth estimates for the analytic solutions of this equation if the coefficients satisfy some conditions.

Cited by 5

Generalized Cesàro-like operator from a class of analytic function spaces to analytic Besov spaces

Pengcheng Tang · arXiv (Cornell University) · 2023

Let $μ$ be a finite positive Borel measure on $[0,1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. For $0<α<\infty$, the generalized Cesàro-like operator $\mathcal{C}_{μ,α}$ is defined by $$ \mathcal {C}_{μ,α}(f)(z)=\sum^\infty_{n=0}\left(μ_n\sum^n_{k=0}\frac{Γ(n-k+α)}{Γ(α)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $μ_n$ denotes the $n$-th moment of the measure $μ$, that is, $μ_n=\int_{0}^{1} t^{n}dμ(t)$. For $s>1$, let $X$ be a Banach subspace of $ H(\mathbb{D})$ with $Λ^{s}_{\frac{1}{s}}\subset X\subset\mathcal {B}$. In this paper, for $1\leq p <\infty$, we characterize the measure $μ$ for which $\mathcal{C}_{μ,α}$ is bounded(or compact) from $X$ into analytic Besov space $B_{p}$.

Cited by 0Open access

A Cesàro-like operator from a class of analytic function spaces to analytic Besov spaces

Pengcheng Tang · Canadian Mathematical Bulletin · 2025

Abstract Let $\mu $ be a finite positive Borel measure on $[0,1)$ and $f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} \in H(\mathbb {D})$ . For $0<\alpha <\infty $ , the generalized Cesàro-like operator $\mathcal {C}_{\mu ,\alpha }$ is defined by $$ \begin{align*}\mathcal {C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n, \ z\in \mathbb{D}, \end{align*} $$ where, for $n\geq 0$ , $\mu _n$ denotes the n th moment of the measure $\mu $ , that is, $\mu _n=\int _{0}^{1} t^{n}d\mu (t)$ . For $s>1$ , let X be a Banach subspace of $H(\mathbb {D})$ with $\Lambda ^{s}_{\frac {1}{s}}\subset X\subset \mathcal {B}$ . In this article, for $1\leq p <\infty $ , we characterize the measure $\mu $ for which $\mathcal {C}_{\mu ,\alpha }$ is bounded (resp. compact) from X into the analytic Besov space $B_{p}$ .

Cited by 0Open access

Volterra type integral operator and analytic function spaces

Rahim Kargar · arXiv (Cornell University) · 2018

We investigate the geometric properties of the Volterra-type integral operator \begin{equation*} T_g[f](z) = \int_{0}^{z} f(s)\, g'(s)\, ds, \quad |z|<1, \end{equation*} acting on various subclasses of analytic functions in the unit disk. Sharp estimates are obtained for the convexity radius of $T_g$, which simultaneously determine its univalence radius, across several classical function families. In addition, we introduce and study higher-order Volterra-type operators, establish their normalized forms, and propose an open question on the scaling behavior of their convexity radii.

Cited by 0Open access

Mixed norms and analytic function spaces
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Buckley, Stephen M. · Maynooth University ePrints and eTheses Archive (Maynooth University) · 2000

We dene and investigate general mixed-norm type sequence spaces, and strengthen inequalities of Hardy-Littlewood, Hausdor-Young, and others.

Cited by 0Open access

Sharp restrictions of analytic function spaces of several variables

Shamoyan, R. F., Makhina, N. M. · ArXiv.org · 2025

In this expository paper we collect many recent advances in analytic function spaces of several complex variables related with trace problem in tubular domains over symmetric cones and bounded strongly pseudoconvex domains with smooth boundary. We consider various function space of analytic functions of several variables in various domains in $C^n$ and provide or complete descriptions of traces or estimates of traces of various analytic function spaces in various domains obtained in recent years, by various authors. The problem to find sharp estimates of traces of Hardy analytic function spaces in the unit polydisk first was posed by W. Rudin in 1969. Since then many papers appeared in literature. We collect in this expository paper not only already many known results on traces of various analytic function spaces in product domains but also discuss various new interesting results related with this problem. Related to trace problem various results were provided previously by G. Henkin, E. Amar, H. Alexander and various other authors. Finnaly, note that our trace theorems are closely related with the Bergman type projections acting between function spaces with different dimensions. This expository paper contains mainly new results concerning traces in tubular and bounded strongly pseudoconvex domains, proofs of these theorems are based in particular also on various properties of Bergman type projection, in this expository paper we will also shortly discuss some new results obtained by first author on Bergman type projections in these complicated domains in $C^n$. This is the second part of our notes related with trace problem. In the first part we provided a large list of recent sharp results on traces in the polydisk and polyball and was published in [1].

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Integral Superposition-Type Operators on Some Analytic Function Spaces

A‎. ‎El-Sayed Ahmed, Saleh Omran · Journal of Function Spaces · 2021

All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the so-called <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>n</mi> </math> -generalized superposition operator are characterized in this paper. Moreover, certain specific properties such as boundedness and compactness of the newly defined class of generalized integral superposition operators are discussed and established by using the concerned entire functions.

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