Cesaro operators on the Hardy spaces of the half-plane

摘要

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 companionoperator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify$\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups ofcomposition operators and we find the norm and the spectrum in each case. Therelation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}rooperators on Lebesgue spaces $L^p(\R)$ of the boundary line is also discussed.