On Some Applications of a Special Integrodifferential Operators
Let 𝐶(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(𝑥,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={𝑧∈ℂ∶|𝑧|<1} is the unit disc in the complex plane ℂ, and have 𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐶𝐴(𝑛)=𝐶𝐴(𝑛)(𝔻×𝔻) denote the subspa...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/894527 |
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| author | Suna Saltan Yasemin Özel |
| author_facet | Suna Saltan Yasemin Özel |
| author_sort | Suna Saltan |
| collection | DOAJ |
| description | Let 𝐶(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(𝑥,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={𝑧∈ℂ∶|𝑧|<1} is the unit disc in the complex plane
ℂ, and have 𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐶𝐴(𝑛)=𝐶𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐶(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐶𝐴(𝑛)=𝐶(𝑛)(𝔻×𝔻)∩ℋ𝑜𝑙(𝔻×𝔻)). The double integration operator is defined in 𝐶𝐴(𝑛) by the formula ∫𝑊𝑓(𝑧,𝑤)=𝑧0∫𝑤0𝑓(𝑢,𝑣)𝑑𝑣𝑑𝑢. By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator 𝑊∣𝐸𝑧𝑤, where 𝐸𝑧𝑤={𝑓∈𝐶𝐴(𝑛)∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤)} is an invariant subspace of 𝑊, and study its properties. We also study invertibility of the elements in 𝐶𝐴(𝑛) with respect to the Duhamel product. |
| format | Article |
| id | doaj-art-a9f2937ec5d540e78cb7876d1115b843 |
| institution | Kabale University |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
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| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-a9f2937ec5d540e78cb7876d1115b8432025-02-03T05:47:42ZengWileyJournal of Function Spaces and Applications0972-68021758-49652012-01-01201210.1155/2012/894527894527On Some Applications of a Special Integrodifferential OperatorsSuna Saltan0Yasemin Özel1Department of Mathematics, Suleyman Demirel University, 32260 Isparta, TurkeyDepartment of Mathematics, Suleyman Demirel University, 32260 Isparta, TurkeyLet 𝐶(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(𝑥,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={𝑧∈ℂ∶|𝑧|<1} is the unit disc in the complex plane ℂ, and have 𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐶𝐴(𝑛)=𝐶𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐶(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐶𝐴(𝑛)=𝐶(𝑛)(𝔻×𝔻)∩ℋ𝑜𝑙(𝔻×𝔻)). The double integration operator is defined in 𝐶𝐴(𝑛) by the formula ∫𝑊𝑓(𝑧,𝑤)=𝑧0∫𝑤0𝑓(𝑢,𝑣)𝑑𝑣𝑑𝑢. By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator 𝑊∣𝐸𝑧𝑤, where 𝐸𝑧𝑤={𝑓∈𝐶𝐴(𝑛)∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤)} is an invariant subspace of 𝑊, and study its properties. We also study invertibility of the elements in 𝐶𝐴(𝑛) with respect to the Duhamel product.http://dx.doi.org/10.1155/2012/894527 |
| spellingShingle | Suna Saltan Yasemin Özel On Some Applications of a Special Integrodifferential Operators Journal of Function Spaces and Applications |
| title | On Some Applications of a Special Integrodifferential Operators |
| title_full | On Some Applications of a Special Integrodifferential Operators |
| title_fullStr | On Some Applications of a Special Integrodifferential Operators |
| title_full_unstemmed | On Some Applications of a Special Integrodifferential Operators |
| title_short | On Some Applications of a Special Integrodifferential Operators |
| title_sort | on some applications of a special integrodifferential operators |
| url | http://dx.doi.org/10.1155/2012/894527 |
| work_keys_str_mv | AT sunasaltan onsomeapplicationsofaspecialintegrodifferentialoperators AT yaseminozel onsomeapplicationsofaspecialintegrodifferentialoperators |