Hardy Spaces of Dirichlet Series and Their Composition Operators

Abstract. Let H 2 denote the space of ordinary Dirichlet series with square summable coefficients, and let H 2 0 denote its subspace consisting of series vanishing at +∞. We investigate the weak product spaces H 2 ⊙ H 2 and H 2 0 ⊙ H 2 0 , finding that several pertinent problems are more tractable for the latter space. This surprising phenomenon is related to the fact thatdoes not contain the infinite-dimensional subspace of H 2 of series which lift to linear functions on the infinite polydisc. The problems considered stem from questions about the dual spaces of these weak product spaces, and are therefore naturally phrased in terms of multiplicative Hankel forms. We show that there are bounded, even Schatten class, multiplicative Hankel forms on H 2 0 × H 2 0 whose analytic symbols are not in H 2 . Based on this result we examine Nehari's theorem for such Hankel forms. We define also the skew product spaces associated with H 2 ⊙ H 2 and H 2 0 ⊙ H 2 0 , with respect to both half-plane and polydisc differentiation, the latter arising from Bohr's point of view. In the process we supply square function characterizations of the Hardy spaces H p , for 0 < p < ∞, from the viewpoints of both types of differentiation. Finally we compare the skew product spaces to the weak product spaces, leading naturally to an interesting Schur multiplier problem.

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“…We will require the following basic result, which can be extracted from Lemma 5 and Theorem 5 in [4]. Lemma 7.…”

mentioning

confidence: 99%

Weak Product Spaces of Dirichlet Series

Brevig

Perfekt

2016

Integr. Equ. Oper. Theory

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“…We know by construction that it is possible to build universal Dirichlet series j≥1 a j j −s satisfying a j = o(j −r ) for any r < 1 (see [3]). Hence, in contrast to the analytic case [7], we have W a ∩ H 2 = ∅ where H 2 is the analogue of H 2 (D) for Dirichlet series (see for instance [2]). …”

Section: O Demanze and A Mouzementioning

confidence: 97%

Growth of coefficients of universal Dirichlet series

Demanze

Mouze

2007

Ann. Polon. Math.

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“…Si au contraire un opérateur T en dimension un a une norme ≤ 1, sa tensorisation est souvent possible et se révèle très utile (voir [9] et [29] par exemple).…”

Section: Définition Via Bohrunclassified

“…Ce renforcement de l'énoncé a été obtenu dans [49], et peut être vu comme une conséquence immédiate de la version "partie réelle" du Théorème de Bohr du chapitre 2. Ajoutons une remarque souvent utile dans ce contexte ( [9], Lemme 11) :…”

Section: Théorème Fondamentalunclassified

Espaces de séries de Dirichlet et leurs opérateurs de composition

Queffélec¹

2015

Ann. math. Blaise Pascal

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Hardy Spaces of Dirichlet Series and Their Composition Operators

Cited by 179 publications

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Weak Product Spaces of Dirichlet Series

Weak Product Spaces of Dirichlet Series

Growth of coefficients of universal Dirichlet series

Espaces de séries de Dirichlet et leurs opérateurs de composition

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