On the Ces`aro operator on the Hardy space in the upper half-plane

TL;DR

This paper investigates the Cesàro operator on the Hardy space in the upper half-plane, providing a new proof of its boundedness and revealing its structure as a sum of the identity and a unitary operator, thus establishing its normality.

Contribution

It offers a novel, simplified proof of the Cesàro operator's boundedness and characterizes it as the sum of the identity and a unitary operator, demonstrating its normality.

Findings

Cesàro operator is bounded on the Hardy space in the upper half-plane

The operator equals the sum of the identity and a unitary operator

The Cesàro operator is normal

Abstract

We discuss the Ces`aro operator on the Hardy space in the upper half-plane. We provide a new simple proof of the boundedness of this operator, prove that this operator is equal to the sum of the identity operator and a unitary operator, which implies its normality.