Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series

By Fr'ed'eric Bayart, Hervé Queffélec, Kristian Seip. 2014

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By a theorem of the first named author, φ generates a bounded composition operator on the Hardy space H pof Dirichlet series (1 6 p 1/2 if c0 = 0 and is either identically zero or maps the right half-plane into itself if c0 is positive. It is shown that the nth approximation numbers of bounded composition operators on H p are bounded below by a constant times rn for some 0 0 when c0 is positive. Both results are best possible. Estimates rely on a combination of soft tools from Banach space theory (s-numbers, type and cotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain interpolation method for H 2, developed in an earlier paper, using estimates of solutions of the ∂ equation. A transference principle from Hp of the unit disc is discussed, leading to explicit examples of compact composition operators on H 1 with approximation numbers decaying at a variety of sub-exponential rates. Finally, a new Littlewood–Paley formula is established, yielding a sufficient condition for a composition operator on H p to be compact. Nombres d’approximation des operateurs de composition sur les espaces Hp des series de Dirichlet

Subfields: Applied Mathematics

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