Hilbert-Type Operators Acting on Bergman Spaces. - Aguilar-Hernández, Tanausú; Galanopoulos, Petros; Girela-Álvarez, Daniel (Springer Nature, 2024)
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) = \sum_{k=0}^{∞} a_{k} z^{k} , z ∈ D, H_{µ}(f) is formally defined by H_{µ}(f)(z)=\sum_{n=0}^{∞} ( \sum_{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 ).
This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at https://doi.org/10.1007/s40315-024-00560-5
The following information regarding funding is not present in either the postprint or published version of the article: "The first author was supported in part by Ministerio de Universidades: "Margarita Salas", grant (Funded by the Spanish Recovery, Transformation and Resilience Plan, and European Union - NextGenerationEU), and by a grant from Ministerio de Ciencia e Innovación, Spain (PID2022-136320NB-I00). The third author was supported in part by a grant from Ministerio de Ciencia e Innovación, Spain (PID2022-136619NB-I00) and by a grant from La Junta de Andalucía, Spain (FQM-210)."
Keywords: Hilbert, Operadores en espacio de; Funciones de variable compleja; Operadores, Teoría de; Generalized Hilbert operator; Bergman spaces; Duality; Carleson measures; Bounded operator; Compact operator; Schatten classes; The Hilbert matrix