Hausdorff–Zhu operators on the unit ball in ℂn C#x0005E;n

We introduce and study Hausdorff–Zhu operators over the unit ball in . We give sufficient conditions for boundedness of such operators in Möbius invariant spaces, weighted Lebesgue and Bergman spaces, and Hardy spaces on the unit ball. In the framework of the Lebesgue spaces, we also provide necessity conditions for boundedness. Approximation of the identity by Hausdorff operators is also considered within Lebesgue space with a Möbius invariant Haar measure.