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Cesàro Operators on the Hardy Spaces of the Half-Plane

Published:2011-08-03
Printed: Jun 2013
• Athanasios G. Arvanitidis,
Department of Mathematics, University of Thessaloniki, 54124 Thessaloniki, Greece
• Aristomenis G. Siskakis,
Department of Mathematics, University of Thessaloniki, 54124 Thessaloniki, Greece
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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 Ces\`{a}ro operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also discussed.
 Keywords: Cesàro operators, Hardy spaces, semigroups, composition operators
 MSC Classifications: 47B38 - Operators on function spaces (general) 30H10 - Hardy spaces 47D03 - Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}