Topological and Functional Properties of Some F-Algebras of Holomorphic Functions
Let Np (1<p<∞) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane. The space Np equipped with the topology given by the metric dp defined by dp(f,g)=(∫02π(log(1+|f∗(eiθ)-g∗(eiθ)|))p(dθ/2π))1/p, f,g∈Np, becomes an F-algebra. For each p>1, we also co...
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Wiley
2015-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2015/850709 |
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| author | Romeo Meštrović |
| author_facet | Romeo Meštrović |
| author_sort | Romeo Meštrović |
| collection | DOAJ |
| description | Let Np (1<p<∞) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane. The space Np equipped with the topology given by the metric dp defined by dp(f,g)=(∫02π(log(1+|f∗(eiθ)-g∗(eiθ)|))p(dθ/2π))1/p, f,g∈Np, becomes an F-algebra. For each p>1, we also consider the countably normed Fréchet algebra Fp of holomorphic functions on D which is the Fréchet envelope of the space Np. Notice that the spaces Fp and Np have the same topological duals. In this paper, we give a characterization of bounded subsets of the spaces Fp and weakly bounded subsets of the spaces Np with p>1. If (Fp)∗ denotes the strong dual space of Fp and Npw∗ denotes the space Sp of complex sequences γ={γn}n satisfying the condition γn=Oexp-cn1/(p+1), equipped with the topology of uniform convergence on weakly bounded subsets of Np, then we prove that Fp∗=Npw∗ both set theoretically and topologically. We prove that for each p>1 Fp is a Montel space and that both spaces Fp and (Fp)∗ are reflexive. |
| format | Article |
| id | doaj-art-40a90dc8a9ca4b05ba4075f55f9bd542 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-40a90dc8a9ca4b05ba4075f55f9bd5422025-08-20T03:55:35ZengWileyJournal of Function Spaces2314-88962314-88882015-01-01201510.1155/2015/850709850709Topological and Functional Properties of Some F-Algebras of Holomorphic FunctionsRomeo Meštrović0Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, MontenegroLet Np (1<p<∞) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane. The space Np equipped with the topology given by the metric dp defined by dp(f,g)=(∫02π(log(1+|f∗(eiθ)-g∗(eiθ)|))p(dθ/2π))1/p, f,g∈Np, becomes an F-algebra. For each p>1, we also consider the countably normed Fréchet algebra Fp of holomorphic functions on D which is the Fréchet envelope of the space Np. Notice that the spaces Fp and Np have the same topological duals. In this paper, we give a characterization of bounded subsets of the spaces Fp and weakly bounded subsets of the spaces Np with p>1. If (Fp)∗ denotes the strong dual space of Fp and Npw∗ denotes the space Sp of complex sequences γ={γn}n satisfying the condition γn=Oexp-cn1/(p+1), equipped with the topology of uniform convergence on weakly bounded subsets of Np, then we prove that Fp∗=Npw∗ both set theoretically and topologically. We prove that for each p>1 Fp is a Montel space and that both spaces Fp and (Fp)∗ are reflexive.http://dx.doi.org/10.1155/2015/850709 |
| spellingShingle | Romeo Meštrović Topological and Functional Properties of Some F-Algebras of Holomorphic Functions Journal of Function Spaces |
| title | Topological and Functional Properties of Some F-Algebras of Holomorphic Functions |
| title_full | Topological and Functional Properties of Some F-Algebras of Holomorphic Functions |
| title_fullStr | Topological and Functional Properties of Some F-Algebras of Holomorphic Functions |
| title_full_unstemmed | Topological and Functional Properties of Some F-Algebras of Holomorphic Functions |
| title_short | Topological and Functional Properties of Some F-Algebras of Holomorphic Functions |
| title_sort | topological and functional properties of some f algebras of holomorphic functions |
| url | http://dx.doi.org/10.1155/2015/850709 |
| work_keys_str_mv | AT romeomestrovic topologicalandfunctionalpropertiesofsomefalgebrasofholomorphicfunctions |