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Cesàro Operators on the Hardy Spaces of the Half-Plane |
Canad. Math. Bull. 56(2013), 229-240
Published:2011-08-03
Printed: Jun 2013
Printed: Jun 2013
- Athanasios G. Arvanitidis,
- Aristomenis G. Siskakis,
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} |