Some revisited results about composition operators on Hardy spaces

Lefèvre, Pascal; Li, Daniel Li; Queffélec, Hervé; Rodrı́guez-Piazza, Luis

doi:10.4171/rmi/666

Cited by 32 publications

(38 citation statements)

References 31 publications

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“…Since C ϕ (H ∞ ) ⊆ H ∞ , C ϕ maps HM Ψ into itself and C ϕ : H Ψ → H Ψ being compact, its restriction C ϕ : HM Ψ → HM Ψ is compact too. We know that the evaluation δ a : f ∈ HM Ψ → f (a) ∈ C has norm ≈ Ψ Other consequences will be given in the subsequent paper [10].…”

Section: Some Consequencesmentioning

confidence: 96%

Nevanlinna counting function and Carleson function of analytic maps

Lefèvre

Queffélec

et al. 2010

Math. Ann.

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Section: Some Consequencesmentioning

confidence: 96%

Nevanlinna counting function and Carleson function of analytic maps

Lefèvre

Queffélec

et al. 2010

Math. Ann.

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“…We combine the geometric approach with recent results [16,17] that clarify the connection between the Nevanlinna counting function N ϕ and the measure µ ϕ , in order to obtain the following characterization. The article is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Trace ideal criteria for embeddings and composition operators on model spaces

Aleman¹,

Lyubarskii²,

Malinnikova³

et al. 2016

Journal of Functional Analysis

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Let K ϑ be a model space generated by an inner function ϑ. We study the Schatten class membership of embeddings I : K ϑ ֒→ L 2 (µ), µ a positive measure, and of composition operators Cϕ :In the case of onecomponent inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I and Cϕ to the Hardy-Smirnov space E 2 (D) in some domain D ⊃ D. In particular, we obtain a characterization of Schatten membership of Cϕ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.

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“…As we have already mentioned, when b ∞ < 1, then H (b) = H 2 , with a norm equivalent to the usual H 2 norm. In this situation, Lefèvre et al [22] characterized reverse Carleson measures under the additional assumption that µ is already a Carleson measure. In [19], the authors were able to get rid of this extra assumption using a balayage type argument.…”

Section: Introductionmentioning

confidence: 99%

Direct and reverse Carleson measures for $\mathscr{H}(b)$ spaces

Blandignères¹,

Fricain²,

Gaunard³

et al. 2015

Indiana Univ. Math. J.

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In this paper we discuss direct and reverse Carleson measures for the de Branges-Rovnyak spaces H (b), mainly when b is a non-extreme point of the unit ball of H ∞ .Here b belongs to H ∞ 1 , the unit ball in H ∞ , and H ∞ is the Banach algebra of bounded analytic functions on D normed with the supremum norm · ∞ . The space H (b) is often known as the de Branges-Rovnyak space and we will review the basics of this space in a moment. For now, note that when b ∞ < 1, then H (b) is just the classical Hardy space H 2 [16,18] with an equivalent norm while if b is an inner function, meaning |b| = 1 almost everywhere on T = ∂D, then H (b) is the classical model space (bH 2 ) ⊥ = H 2 ⊖ bH 2 . For any b, the space H (b) is contractively contained in H 2 . As is often the case, the properties of H (b) spaces, including direct and reverse Carleson measures, depend on whether or not b is an extreme point ofwhere m is normalized Lebesgue measure on the unit circle T.Let M + (D − ) denote the positive finite Borel measures on the closed unit disk D − . By a reverse Carleson measure for H (b) we mean a 2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.

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Some revisited results about composition operators on Hardy spaces

Cited by 32 publications

References 31 publications

Nevanlinna counting function and Carleson function of analytic maps

Nevanlinna counting function and Carleson function of analytic maps

Trace ideal criteria for embeddings and composition operators on model spaces

Direct and reverse Carleson measures for $\mathscr{H}(b)$ spaces

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