Composition operators on spaces of double Dirichlet series
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Composition operators on spaces of double Dirichlet series Frédéric Bayart1 · Jaime Castillo-Medina2 Manuel Maestre2 · Pablo Sevilla-Peris3
· Domingo García2 ·
Received: 25 April 2019 / Accepted: 20 December 2019 © Universidad Complutense de Madrid 2020
Abstract We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series H∞ (C2+ ). We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in H∞ (C+ ) and in the spaces H p . Keywords Double Dirichlet series · Composition operator · Superposition operator Mathematics Subject Classification 30B50 · 47B33 · 46J15 · 32A10
1 Introduction The study of composition operators appears as a consistent topic of interest in the literature of Banach spaces of holomorphic functions. Generally, a composition operator is defined by a function φ, called the symbol of the composition operator Cφ , so that Cφ ( f ) = f ◦ φ. Composition operators of spaces of Dirichlet series were first studied in [7], where the authors focus on H2 , the Hilbert space of Dirichlet series whose sequence of coefficients belongs to 2 . Before we continue, let us introduce some notation. We will denote by Cσ , with σ > 0, the half-plane of complex numbers whose real part is strictly larger than σ , using C+ as a substitute notation for the case of C0 to highlight the relevance of this special case. We denote by D the set of
The first author was partially supported by the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). The last four authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second author was also supported by Grant FPU14/04365 and MICINN. The third and fourth authors were also supported by project Prometeo/2017/102 of the Generalitat Valenciana.
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Jaime Castillo-Medina [email protected]
Extended author information available on the last page of the article
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Dirichlet series which converge in some half-plane, that is, the set of Dirichlet series which converge somewhere, hence in a half-plane (see [9, Lemma 4.1.1]). The two main results of [7] (see also [9, Theorem 6.4.5]) characterize composition operators acting on H2 . Theorem 1 [7, Theorem A] Let θ ∈ R. An analytic function φ : Cθ → C 1 generates 2
a composition operator Cφ : H2 → D if and only if it is of the form φ(s) = c0 s +ϕ(s) with c0 ∈ N0 and ϕ ∈ D. Theorem 2 [7, Theorem B] An analytic function φ : C 1 → C 1 defines a bounded composition operator Cφ : H2 → H2 if and only if
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(a) it is of the form φ(s) = c0 s + ϕ(s) with c0 ∈ N0 and ϕ ∈ D. (b) φ has an analytic extension to C+ , also denoted by φ, such that (i) φ(C+ ) ⊂ C+ i f c0 > 0, and (ii) φ(C+ ) ⊂ C 1 i f c0 = 0. 2
Here it should be understood that ϕ is a Dirichlet series which
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