Hilbert-Type Operators Acting on Bergman Spaces.

Abstract

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 ).