The differentiation operator in the space of uniformly convergent Dirichlet series

Bonet, José

doi:10.1002/mana.201900211

Cited by 2 publications

(2 citation statements)

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“…Also (24) shows that the inverse (λI − D) −1 is continuous, giving our claim. The rest of the proof follows exactly as that of [Bon,Theorem 2.6]. P. Sevilla-Peris Insitut Universitari de Matemàtica Pura i Aplicada.…”

Section: σ(D Hmentioning

confidence: 74%

“…We finish this note by looking at the spectrum of the differentiation and integration operators, in the same spirit as [Bon,Theorem 2.6]. Let us recall that the resolvent of a linear operator T : X → X (where X is some Fréchet space) is defined as the set ρ(T, X ) consisting of those λ ∈ C for which (λI −T ) is bijective and its inverse is continuous.…”

Section: Differentiation and Integration Operatorsmentioning

confidence: 99%

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Hardy space of translated Dirichlet series

et al. 2021

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We study the Hardy space of translated Dirichlet series H + . It consists on those Dirichlet series a n n −s such that for some (equivalently, every) 1 ≤ p < ∞, the translation a n n −(s+ 1 σ ) belongs to the Hardy space H p for every σ > 0. We prove that this set, endowed with the topology induced by the seminorms • 2,k k∈N (whereH 2 ), is a Fréchet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials {n −s } n∈N are an unconditional Schauder basis of H + . We also explore the connection of this new space with spaces of holomorphic functions on infinite-dimensional spaces.In the spirit of Gordon and Hedenmalm's work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on H + and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces.

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Section: σ(D Hmentioning

confidence: 74%

Section: Differentiation and Integration Operatorsmentioning

confidence: 99%