Composition operators on a Hilbert space of Dirichlet Series

Catherine Finet

Université de Mons-Hainaut, Institut de Mathématique, Avenue du Champ de Mars, 6, B7000 Mons (Belgium)

We study the numerical range and compacity of some composition operators, defined on a Hilbert space of Dirichlet series introduced by Hedenmalm, Linqvist and Seip. We describe the numerical range of “nice” composition operators. We focus on the zero-inclusion question. The compacity is studied for symbols

ϕ(s) = c_{0}s + c_{1} + ∑_{j=1}^{d} c_{q_{j}}q_{j}^{−s} (d = 1, 2, 3)

where the q_{j}’s are multiplicatively independent. The situation is completely different, according whether c_{0} ≥ 1 or c_{0} = 0. This is a joint work with H. Quefflec and A. Volberg.