Hilbert-Type Operators Acting on Bergman Spaces Hilbert-Type Operators Acting on Bergman Spaces

If μ is a positive Borel measure on the interval [0, 1) we let H μ be the Hankel matrix H μ = ( μ n , k ) n , k ≥ 0 with entries μ n , k = μ n + k , where, for n = 0 , 1 , 2 , … , μ n denotes the moment of order n of μ . This matrix formally induces an operator, called also H μ , on the space of all...

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Bibliographic Details
Published in:	Computational methods and function theory Vol. 25; no. 4; pp. 863 - 888
Main Authors:	Aguilar-Hernández, Tanausú, Galanopoulos, Petros, Girela, Daniel
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2025 Springer Nature B.V
Subjects:	Analysis Analytic functions Computational Mathematics and Numerical Analysis Functions of a Complex Variable Hankel matrices Mathematical analysis Mathematical functions Mathematics Mathematics and Statistics Operators (mathematics) Compact operator Schatten classes The Hilbert matrix 47B35 Bounded operator Generalized Hilbert operator Duality Bergman spaces Carleson measures 30H20
ISSN:	1617-9447, 2195-3724
Online Access:	Get full text
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Summary:	If μ is a positive Borel measure on the interval [0, 1) we let H μ be the Hankel matrix H μ = ( μ n , k ) n , k ≥ 0 with entries μ n , k = μ n + k , where, for n = 0 , 1 , 2 , … , μ n denotes the moment of order n of μ . This matrix formally induces an operator, called also H μ , on the space of all analytic functions in the unit disc D as follows: If f is an analytic function in D , f ( z ) = ∑ k = 0 ∞ a k z k , z ∈ D , H μ ( f ) is formally defined by H μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n + k a k z n , z ∈ D . This is a natural generalization of the classical Hilbert operator. This paper is devoted to studying the operators H μ acting on the Bergman spaces A p , 1 ≤ p < ∞ . Among other results, we give a complete characterization of those μ for which H μ 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 H μ on A p for the other values of p , as well as on its membership in the Schatten classes S p ( A 2 ) .
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1617-9447 2195-3724
DOI:	10.1007/s40315-024-00560-5