Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

Bonet, José

doi:10.1007/s13398-022-01323-4

Cited by 6 publications

(7 citation statements)

References 152 publications

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“…lim t→+∞ t k v(t) = 0 for all k ≥ 0. This is the same notion to be a weight (function) as it has been considered in [21] and several other recent papers; see [6], [4], [19], and the informative survey article [2]. It is known that H ∞ v (C) is a Banach space w.r.t.…”

Section: Introductionmentioning

confidence: 92%

“…for all ǫ > 0 we can find a (radial symmetric) compact set K ⊆ C such that |f (z)|v(|z|) < ǫ for all z ∈ C\K. In [2] these classes are denoted by H 0 v (C) and similarly H o v (D) can be introduced; see e.g. [3].…”

Section: Weighted Classes Of Entire Functions With O-growth Restrictionmentioning

confidence: 99%

“…for the first estimate the second part of (4.13) has been used. Note that M (f 2 v N c , t) = M (f v N c , t) 2 . Let now d ∈ N >0 be given and set c ′ = 1 2d .…”

Section: 3mentioning

confidence: 99%

“…The author thanks Prof. José Bonet Solves from the Universitat Politècnica de València for helpful and clarifying explanations during the preparation of this work. In particular, for forwarding the recent article [2]. Moreover, the author thanks Armin Rainer from the University of Vienna for his help and joint discussions concerning Appendix A.…”

Section: Introductionmentioning

confidence: 95%

“…Note that relation ∼ is denoted by the symbol ≈ in [2] and by ≍ in [8] and [9]. In the central statements [21,Thm.…”

Section: Introductionmentioning

confidence: 99%

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2020

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We characterize the inclusion relations between weighted classes of entire functions with rapid decreasing growth and study strong growth comparison relations between given weights. In our considerations first we focus on weights defined in terms of the so-called associated weight function where the weight(system) is based on a given sequence. Then the abstract weight function case is reduced to the weight sequence setting by using the so-called associated weight sequence. Finally, we compare weighted entire function spaces defined in terms of so-called dilatation-type and exponential-type weight systems.

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Section: Introductionmentioning

confidence: 92%

Section: Weighted Classes Of Entire Functions With O-growth Restrictionmentioning

confidence: 99%

“…for the first estimate the second part of (4.13) has been used. Note that M (f 2 v N c , t) = M (f v N c , t) 2 . Let now d ∈ N >0 be given and set c ′ = 1 2d .…”

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

“…Note that relation ∼ is denoted by the symbol ≈ in [2] and by ≍ in [8] and [9]. In the central statements [21,Thm.…”

Section: Introductionmentioning

confidence: 99%

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2020

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The Bishop–Phelps–Bollobás Property for Weighted Holomorphic Mappings

Jiménez-Vargas,

Ramírez,

Villegas-Vallecillos

2024

Results Math

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Given an open subset U of a complex Banach space E, a weight v on U and a complex Banach space F, let $$H^\infty _v(U,F)$$ H v ∞ ( U , F ) denote the Banach space of all weighted holomorphic mappings from U into F, endowed with the weighted supremum norm. We introduce and study a version of the Bishop–Phelps–Bollobás property for $$H^\infty _v(U,F)$$ H v ∞ ( U , F ) ($$WH^\infty $$ W H ∞ -BPB property, for short). A result of Lindenstrauss type with sufficient conditions for $$H^\infty _v(U,F)$$ H v ∞ ( U , F ) to have the $$WH^\infty $$ W H ∞ -BPB property for every space F is stated. This is the case of $$H^\infty _{v_p}(\mathbb {D},F)$$ H v p ∞ ( D , F ) with $$p\ge 1$$ p ≥ 1 , where $$v_p$$ v p is the standard polynomial weight on $$\mathbb {D}$$ D . The study of the relations of the $$WH^\infty $$ W H ∞ -BPB property for the complex and vector-valued cases is also addressed as well as the extension of the cited property for mappings $$f\in H^\infty _v(U,F)$$ f ∈ H v ∞ ( U , F ) such that vf has a relatively compact range in F.

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Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

Albanese,

Bonet,

Ricker

2024

Collect. Math.

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An investigation is made of the generalized Cesàro operators $$C_t$$ C t , for $$t\in [0,1]$$ t ∈ [ 0 , 1 ] , when they act on the space $$H({{\mathbb {D}}})$$ H ( D ) of holomorphic functions on the open unit disc $${{\mathbb {D}}}$$ D , on the Banach space $$H^\infty $$ H ∞ of bounded analytic functions and on the weighted Banach spaces $$H_v^\infty $$ H v ∞ and $$H_v^0$$ H v 0 with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of $$C_t$$ C t as well as their linear dynamics and mean ergodicity.

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Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

Cited by 6 publications

References 152 publications

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The Bishop–Phelps–Bollobás Property for Weighted Holomorphic Mappings

Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

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