作用于伯格曼空间的希尔伯特型算子

IF 0.6 4区 数学 Q3 MATHEMATICS Computational Methods and Function Theory Pub Date : 2024-09-02 DOI:10.1007/s40315-024-00560-5
Tanausú Aguilar-Hernández, Petros Galanopoulos, Daniel Girela
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引用次数: 0

摘要

如果 \(\mu \)是区间[0, 1]上的正博尔量纲,我们让 \({\mathcal {H}}_\mu \)是汉克尔矩阵 \({\mathcal {H}}_\mu =(\mu _{n、k})_{n,k\ge 0}\),其中,对于 \(n\,=\,0,1,2,\ldots\),\(\mu _n\)表示\(\mu \)的n阶矩。这个矩阵在单位圆盘中所有解析函数的空间上形式上诱导了一个算子,也叫做 \({\mathcal {H}}_\mu \),如下所示:If f is an analytic function in \({\mathbb {D}}\), \(f(z)=sum _{k=0}^\infty a_kz^k\), \(z\in {{\mathbb {D}}\)、\({\mathcal {H}}_\mu (f)\) 的正式定义是 $$\begin{aligned} {\mathcal {H}}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n+k}{a_k}\right) z^n,\quad zin {\mathbb {D}}.\end{aligned}$$这是经典希尔伯特算子的自然广义化。本文致力于研究作用于伯格曼空间(A^p\ )、(1\le p<\infty \)的算子(H_\mu \)。在其他结果中,我们给出了当p为1或大于2时,\({\mathcal {H}}_\mu \)在空间\(A^p\)上是有界或紧凑的那些\(\mu \)的完整特征。我们还给出了一些关于其他 p 值时 \(\mathcal H_\mu \) 在 \(A^p\) 上的有界性和紧凑性的结果,以及关于它在 Schatten 类 \({mathcal {S}}_p(A^2)\) 中的成员资格的结果。
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Hilbert-Type Operators Acting on Bergman Spaces

If \(\mu \) is a positive Borel measure on the interval [0, 1) we let \({\mathcal {H}}_\mu \) be the Hankel matrix \({\mathcal {H}}_\mu =(\mu _{n, k})_{n,k\ge 0}\) with entries \(\mu _{n, k}=\mu _{n+k}\), where, for \(n\,=\,0, 1, 2, \ldots \), \(\mu _n\) denotes the moment of order n of \(\mu \). This matrix formally induces an operator, called also \({\mathcal {H}}_\mu \), on the space of all analytic functions in the unit disc \({\mathbb {D}}\) as follows: If f is an analytic function in \({\mathbb {D}}\), \(f(z)=\sum _{k=0}^\infty a_kz^k\), \(z\in {{\mathbb {D}}}\), \({\mathcal {H}}_\mu (f)\) is formally defined by

$$\begin{aligned} {\mathcal {H}}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n+k}{a_k}\right) z^n,\quad z\in {\mathbb {D}}. \end{aligned}$$

This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators \(H_\mu \) acting on the Bergman spaces \(A^p\), \(1\le p<\infty \). Among other results, we give a complete characterization of those \(\mu \) for which \({\mathcal {H}}_\mu \) is bounded or compact on the space \(A^p\) when p is either 1 or greater than 2. We also give a number of results concerning the boundedness and compactness of \(\mathcal H_\mu \) on \(A^p\) for the other values of p, as well as on its membership in the Schatten classes \({\mathcal {S}}_p(A^2)\).

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来源期刊
Computational Methods and Function Theory
Computational Methods and Function Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.20
自引率
0.00%
发文量
44
审稿时长
>12 weeks
期刊介绍: CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.
期刊最新文献
On Uniformity Exponents of $$\varphi $$ -Uniform Domains Hilbert-Type Operators Acting on Bergman Spaces A Characterization of Concave Mappings Using the Carathéodory Class and Schwarzian Derivative The $$*$$ -Exponential as a Covering Map Entire Solutions of Certain Type Binomial Differential Equations
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