The Corona Property in Nevanlinna quotient algebras and interpolating sequences

Massaneda, Xavier; Nicolau, Artur; Thomas, Pascal J.

doi:10.1016/j.jfa.2018.08.001

Cited by 6 publications

(15 citation statements)

References 18 publications

(40 reference statements)

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“…The proof of Theorem 1.4 uses the following variant of Lemma 1.1 of [8]. Proof We can assume that H(z) ≥ 100.…”

Section: Claim 2 H ∈ H (B)mentioning

confidence: 99%

“…The analogue of the WEP in the Nevanlinna class was introduced in [8] where it was proved that invertible classes [ f ] in N BN are precisely the classes for which there exists…”

Section: Introductionmentioning

confidence: 99%

“…When I is a Blaschke product, H QB (I) is the whole cone of positive quasiharmonic functions if and only if the zeros of I are a finite union of interpolating sequences in the Smirnov class. See [8]. We have not explored the analogues of our Theorems 1.2-1.5 for the class H QB (B).…”

mentioning

confidence: 99%

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Invertibility Threshold for Nevanlinna Quotient Algebras

Nicolau

Thomas²

2021

Can. J. Math.-J. Can. Math.

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Let N be the Nevanlinna class and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra N/BN, that is, | f | ≥ e −H on the set B −1 {0} for some positive harmonic function H, holds if and only if the function − log |B | has a harmonic majorant on the set {z ∈ D : ρ(z, Λ) ≥ e −H (z) }; at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.

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“…The proof of Theorem 1.4 uses the following variant of Lemma 1.1 of [8]. Proof We can assume that H(z) ≥ 100.…”

Section: Claim 2 H ∈ H (B)mentioning

confidence: 99%

“…The analogue of the WEP in the Nevanlinna class was introduced in [8] where it was proved that invertible classes [ f ] in N BN are precisely the classes for which there exists…”

Section: Introductionmentioning

confidence: 99%

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Invertibility Threshold for Nevanlinna Quotient Algebras

Nicolau

Thomas²

2021

Can. J. Math.-J. Can. Math.

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“…By recent advances concerning free interpolation in N [18,19,32], there is an astounding resemblance between uniformly separated sequences and Nevanlinna interpolating sequences. Therefore the following results can be interpreted as Nevanlinna analogues of ones that are either presented in Sect.…”

Section: Nevanlinna Interpolating Sequencesmentioning

confidence: 99%

“…The interpolation property (30) guarantees that every point z n ∈ Λ is a removable singularity for A. It remains to show that there exists h ∈ Har (29) implies that the two right-most terms in (32) are of the desired type. Since B + 2B (B g + Bg ) vanishes on the sequence Λ, Lemma 3 shows that there exists h 2 ∈ Har…”

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confidence: 99%

Converse growth estimates for ODEs with slowly growing solutions

Gröhn

2020

Math. Z.

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Let $$f_1,f_2$$ f 1 , f 2 be linearly independent solutions of $$f''+Af=0$$ f ′ ′ + A f = 0 , where the coefficient A is an analytic function in the open unit disc $${\mathbb {D}}$$ D of the complex plane $${\mathbb {C}}$$ C . It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function $$u=-\log \, (f_1/f_2)^{\#}$$ u = - log ( f 1 / f 2 ) # . For example, the case when $$\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 < \infty $$ sup z ∈ D | A ( z ) | ( 1 - | z | 2 ) 2 < ∞ and $$f_1/f_2$$ f 1 / f 2 is normal, is characterized by the condition $$\sup _{z\in {\mathbb {D}}} |\nabla u(z)|(1-|z|^2) < \infty $$ sup z ∈ D | ∇ u ( z ) | ( 1 - | z | 2 ) < ∞ . Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u. Even if $$f_1,f_2$$ f 1 , f 2 are bounded linearly independent solutions of $$f''+Af=0$$ f ′ ′ + A f = 0 , it is possible that $$\sup _{z\in {\mathbb {D}}} |A(z)|(1-|z|^2)^2 = \infty $$ sup z ∈ D | A ( z ) | ( 1 - | z | 2 ) 2 = ∞ or $$f_1/f_2$$ f 1 / f 2 is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of $$f''+Af=0$$ f ′ ′ + A f = 0 in such a way that it solves an interpolation problem natural to bounded analytic functions, while $$|A(z)|^2(1-|z|^2)^3\, dm(z)$$ | A ( z ) | 2 ( 1 - | z | 2 ) 3 d m ( z ) remains to be a Carleson measure.

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From ℋ^∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class

Massaneda

Thomas

2020

Concrete Operators

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This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.

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The Corona Property in Nevanlinna quotient algebras and interpolating sequences

Cited by 6 publications

References 18 publications

Invertibility Threshold for Nevanlinna Quotient Algebras

Invertibility Threshold for Nevanlinna Quotient Algebras

Converse growth estimates for ODEs with slowly growing solutions

From ℋ^∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class

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The Corona Property in Nevanlinna quotient algebras and interpolating sequences

Cited by 6 publications

References 18 publications

Invertibility Threshold for Nevanlinna Quotient Algebras

Invertibility Threshold for Nevanlinna Quotient Algebras

Converse growth estimates for ODEs with slowly growing solutions

From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class

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Product

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From ℋ^∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class