Compact composition operators with nonlinear symbols on the H2 space of Dirichlet series

Bayart, Frédéric; Brevig, Ole Fredrik

doi:10.2140/pjm.2017.291.81

Cited by 7 publications

(18 citation statements)

References 11 publications

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“…Fixing for a moment z ∈ T ∞ , we notice that Φ(w) = (Bϕ) w (z) maps D to C 1/2 with Φ(0) = c 1 . Considering therefore P a member of H p i , we apply (5) and conclude that…”

Section: Composition Operators and The Embedding Problemmentioning

confidence: 99%

“…, p d ). This lead us to introduce the notion of minimal Bohr lift in [5]. For the Bergman spaces, we are by definition required to consider the canonical Bohr lift, since it is used to compute the norm.…”

Section: Proof Of Theorem 17 -(Ii) and (Iii)mentioning

confidence: 99%

“…Remark. It is possible to generate more examples from the results in [5] or from the results of Section 4 in combination with Theorem 1. If 2k < p < 2k + 2, we can choose any Dirichlet polynomial with κ ≥ p/2k, where κ as defined in [5,Lem.…”

Section: Composition Operators With Linear Symbols On H Pmentioning

confidence: 99%

“…The exponent 2 p − 2 is the smallest possible. We then perform a careful study of composition operators with polynomial symbols mapping D α to D β , in the spirit of [5]. We show that for certain polynomial symbols, C ϕ maps D α into D β with β < 2 α − 1 and that the optimality of β = 2 α − 1 also can be decided by investigating the most simple non-trivial symbol, namely ϕ(s) = 3/2 − 2 −s .…”

Section: Introductionmentioning

confidence: 99%

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Composition operators and embedding theorems for some function spaces of Dirichlet series

Bayart

Brevig

2019

Math. Z.

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We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols ϕ on a scale of Bergman-type Hilbert spaces D α . We investigate the optimal β such that the composition operator C ϕ maps D α boundedly into D β . We also prove a new embedding theorem for the non-Hilbertian Hardy space H p into a Bergman space in the half-plane and use it to consider composition operators generated by polynomial symbols on H p , finding the first nontrivial results of this type. The embedding also yields a new result for the functional associated to the multiplicative Hilbert matrix. 1 2 < ∞.Here d (n) denotes the number of divisors of the positive integer n. Note that D 0 = H 2 . We are interested in the range α ≥ 0 and, as explained in [1], these spaces may be thought of as Dirichlet series analogues of the classical scale of weighted Bergman spaces in the unit disc. Since d (n) = O (n ε ) for every ε > 0, it follows from the Cauchy-Schwarz inequality that Dirichlet series in D α also are absolutely convergent in C 1/2 .Due to an insight of H. Bohr (see Section 2), both H p and D α can be identified with certain function spaces in countably infinite number of complex variables, and -consequently -the

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“…Fixing for a moment z ∈ T ∞ , we notice that Φ(w) = (Bϕ) w (z) maps D to C 1/2 with Φ(0) = c 1 . Considering therefore P a member of H p i , we apply (5) and conclude that…”

Section: Composition Operators and The Embedding Problemmentioning

confidence: 99%

Section: Proof Of Theorem 17 -(Ii) and (Iii)mentioning

confidence: 99%

Section: Composition Operators With Linear Symbols On H Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Composition operators and embedding theorems for some function spaces of Dirichlet series

Bayart

Brevig

2019

Math. Z.

Self Cite

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“…The beautiful pioneering contribution of Gordon and Hedenmalm [42] and a growing number of other papers have established the study of composition operators on Hardy spaces of Dirichlet series as an active research area in the interface of one and several complex variables. In the series of papers [77,16,12], quantitative and functional analytic tools have been developed in this context, for example norm estimates for linear combinations of reproducing kernels, Littlewood-Paley formulas, and (soft) functional analytic remedies for the fact that H p fails to be complemented when 1 ≤ p < ∞ and p = 2.…”

Section: Operator Theory and Harmonic Analysismentioning

confidence: 99%