Hausdorff Operators on Some Spaces of Holomorphic Functions on the Unit Disc

TL;DR

This paper introduces Hausdorff operators on the unit disc and establishes conditions for their boundedness across various holomorphic function spaces, also exploring their approximation properties.

Contribution

It provides new boundedness criteria for Hausdorff operators on Bloch, Bergman, and Hardy spaces, and investigates their role in identity approximation.

Findings

Boundedness conditions for Hausdorff operators established.

Hausdorff operators shown to approximate identity in certain spaces.

Applicability to multiple classical holomorphic function spaces.

Abstract

We introduce Hausdorff operators over the unit disc and give conditions for boundedness of such operator in Bloch, Bergman, and Hardy spaces on the disc. Identity approximation by Hausdorff operators is also considered.