Cesaro operators on the Hardy spaces of the half-plane

TL;DR

This paper investigates Cesàro operators on Hardy spaces of the upper half-plane, identifying their spectral properties, norms, and connections to semigroup resolvents and boundary Lebesgue spaces.

Contribution

It characterizes the Cesàro operators as resolvents of semigroups, computes their norms and spectra, and relates them to boundary Lebesgue space operators.

Findings

Identified Cesàro operators as resolvents of semigroup generators.

Computed the norm and spectrum of Cesàro operators on Hardy spaces.

Established connections between Hardy space operators and boundary Lebesgue space operators.

Abstract

In this article we study the Ces\`{a}ro operator $$ \mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta, $$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro operators on Lebesgue spaces $L^p(\R)$ of the boundary line is also discussed.