Composition operators on weighted Bergman spaces of Dirichlet series

Bailleul, Maxime

doi:10.1016/j.jmaa.2015.01.010

Cited by 16 publications

(25 citation statements)

References 17 publications

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“…Assume that w n 1. There exists a function F ∈ H w such that (1) For almost all χ ∈ T ∞ , F χ converges in C 0 and cannot be analytically continued to any larger domain; (2) For at least one χ ∈ T ∞ , F χ converges in C 1/2 and cannot be analytically continued to any larger domain.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Zhao

2018

Concrete Operators

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The Hilbert spaces H w consisiting of Dirichlet series F (s) = ∞ n=1 a n n −s that satisfty ∞ n=1 |a n | 2 /w n < ∞, with {w n } n of average order log j n (the j -fold logarithm of n), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon-Hedenmalm theorem on such H w from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ(s) = c 0 s + φ(s), where c 0 is a nonnegative integer and φ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c 0 = 0. It is verified for every integer j 1, real α > 0 and {w n } n having average order (log + j n) α , that the composition operators map H w into a scale of H w ′ with w ′ n having average order (log + j +1 n) α . The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.

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Section: Proof Of Theoremmentioning

confidence: 99%

“…The author is supported by the Research Council of Norway grant 227768. 1 We observe from the proof of this extension (see [2,Theorem 1]) that D α are actually mapped into weighted Hilbert spaces that consist of Dirichlet series F (s) = ∞ n=1 a n n −s satisfying…”

Section: Introductionmentioning

confidence: 99%

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Zhao

2018

Concrete Operators

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“…3 is not zero. By way of contradiction, we assume that c 1) log n = 0 for all n > 1 and some ∈ {2, 3}, which implies that −s / ∈ (C Φ H) ⊥ for = 2 or 3. This contradiction implies that c…”

Section: Invariant Subspacesmentioning

confidence: 99%

“…from function theoretic properties of ϕ and vice versa. We refer to monographs by Cowen-MacCluer [4], Shapiro [15] and Zhu [18] for general information of composition operators on the open unit disc D of the complex plane C. Recently, an extensive study of composition operators was carried out on various spaces of Dirichlet series (see [1,2,3,6,12,16]).…”

Section: Introductionmentioning

confidence: 99%

Invariant subspaces of composition operators on a Hilbert space of Dirichlet series

Wang¹,

Yao²

2015

Ann. Funct. Anal.

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In this paper, we study invariant subspaces of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The structure of invariant subspaces of a composition operator is characterized, and the strongly closed algebras generated by some composition operators with irrational symbols are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.

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“…Composition operators acting on H (or other spaces of Dirichlet series) have been extensively studied in many papers in recent years (see [1,2,3,5,11,18]). Particularly, the following result comes from [11,18], which characterizes the bounded composition operators on H. And it is convenient to call ϕ a c 0 -symbol if ϕ : C 1/2 → C 1/2 is analytic and satisfies the below conditions such that C ϕ is bounded on H. Theorem A.…”

Section: Introductionmentioning

confidence: 99%