Boundedness and compactness of Hausdorff operators on Fock spaces

Abstract

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space F α p F^p_\alpha and taking its values into a larger one F α q ,   0 > p ≤ q ≤ ∞ F^q_\alpha ,\ 0 > p \leq q \leq \infty , as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space F α ∞ F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become p p -summing are also included.