Remark Concerning Ces\'aro Operator on the Hardy Space $H^p(\mathbb{C}_+)$ in the Upper Half-Plane

TL;DR

This paper investigates the spectral properties of the Cesàro operator on the Hardy space in the upper half-plane, revealing that its associated operator's norm exceeds one for all p except 2, where it is unitary.

Contribution

It demonstrates that for all p in (1,∞) except p=2, the operator derived from the Cesàro operator has a norm greater than one, extending previous spectral results.

Findings

For p ≠ 2, the operator V has a norm strictly greater than one.

For p=2, the operator V is unitary.

The spectrum of V lies on the unit circle for all p.

Abstract

We consider Ces\'aro operator on the Hardy space $H^p(\mathbb{C}_+)$ in the upper half-plane for $1<p<\infty$. In \cite{AS} it was proved that for all $1<p<\infty$ the spectrum of the operator $V=\frac{2(p-1)}{p}C-I$ is located on the unit circle and in \cite{ABC1} the authors of this note showed that for $p=2$ operator $V$ is unitary. In the present note we show that for $1<p<\infty$, $p\ne 2$, the norm of the operator $V$ is strictly greater than one.