Cesaro operators on the Hardy spaces of the half-plane

Abstract

In this article we study the Cesàro 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 Cesàro operators on Lebesgue spaces $L^p(\mathbb{R})$ of the boundary line is also discussed.

14 pages