Hilbert-Type Operators Acting on Bergman Spaces: Hilbert-Type Operators Acting on Bergman Spaces - Aguilar-Hernández, Tanausú; Galanopoulos, Petros; Girela, Daniel (Computational methods and function theory, 2025)
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 ) .