Generalized Integration Operators on Hardy Spaces

TL;DR

This paper introduces a generalized integration operator on Hardy spaces, analyzes its boundedness and compactness, and provides simplified proofs of existing theorems in the field.

Contribution

It presents a new generalized operator on Hardy spaces and offers streamlined proofs of known results regarding its properties.

Findings

The operator's boundedness conditions are characterized.

The operator's compactness criteria are established.

Simplified proofs of Rätjä and Cohn's results are provided.

Abstract

We introduce a natural generalization of a well studied integration operator acting on the family of Hardy spaces in the unit disc. We study the boundedness and compactness properties of the operator and finally we use these results to give simple proofs of a result of R\"atty\"a and another result by Cohn.