Schur spaces and weighted spaces of type<i>H</i><sup>∞</sup>

Miralles, Alejandro

doi:10.2989/16073606.2012.742248

Quaestiones Mathematicae

2012

DOI: 10.2989/16073606.2012.742248

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Schur spaces and weighted spaces of typeH^∞

Alejandro Miralles¹

Abstract: Abstract. We extend some results related to composition operators on Hv(G) to arbitrary linear operators on Hv 0 (G) and Hv(G). We also give examples of rank-one operators on Hv(G) which cannot be approximated by composition operators.

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2024

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Cited by 2 publications

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Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Beltrán-Meneu,

Bonet,

Jordá

2024

Anal.Math.Phys.

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Let $$\mu $$ μ be a positive finite Borel measure on [0, 1). Cesàro-type operators $$C_{\mu }$$ C μ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that $$C_\mu $$ C μ is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of $$C_\mu $$ C μ on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments $$(\mu _n)_{n\in {\mathbb {N}}_0}$$ ( μ n ) n ∈ N 0 . The continuity, compactness and spectrum of $$C_\mu $$ C μ acting on Fréchet and (LB) Korenblum type spaces are also considered .

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Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Beltrán-Meneu,

Bonet,

Jordá

2024

Anal.Math.Phys.

View full text Add to dashboard Cite

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Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms

Beltrán-Meneu,

Bonet,

Jordá

2024

Banach J. Math. Anal.

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The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $$\mu $$ μ on [0, 1) is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Cesàro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $$\mu $$ μ . In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed.

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Schur spaces and weighted spaces of typeH^∞

Abstract: Abstract. We extend some results related to composition operators on Hv(G) to arbitrary linear operators on Hv 0 (G) and Hv(G). We also give examples of rank-one operators on Hv(G) which cannot be approximated by composition operators.

Cited by 2 publications

References 23 publications

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms

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Schur spaces and weighted spaces of typeH∞

Abstract: Abstract. We extend some results related to composition operators on Hv(G) to arbitrary linear operators on Hv 0 (G) and Hv(G). We also give examples of rank-one operators on Hv(G) which cannot be approximated by composition operators.

Cited by 2 publications

References 23 publications

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms

Contact Info

Product

Resources

About

Schur spaces and weighted spaces of typeH^∞