Cesaro operators on the Hardy spaces of the half-plane

In this article we study the Cesàro operator $\mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(ζ)\,dζ,$ 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 $L^p(\R)$ of the boundary line is also discussed.

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