Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space
Let 𝜑(𝑧)=𝑧𝑛,𝑧∈𝕌, for some positive integer 𝑛 and 𝐶𝜑 the composition operator on the Dirichlet space 𝒟 induced by 𝜑. In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators 𝐶∗𝜑𝐶𝜑,𝐶𝜑𝐶∗𝜑 and self-commutators of 𝐶𝜑, which expose that t...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/618968 |
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| author | A. Abdollahi S. Mehrangiz |
| author_facet | A. Abdollahi S. Mehrangiz |
| author_sort | A. Abdollahi |
| collection | DOAJ |
| description | Let 𝜑(𝑧)=𝑧𝑛,𝑧∈𝕌, for some positive integer 𝑛 and 𝐶𝜑 the composition operator on the Dirichlet space 𝒟 induced by 𝜑. In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators 𝐶∗𝜑𝐶𝜑,𝐶𝜑𝐶∗𝜑 and self-commutators of 𝐶𝜑, which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators. |
| format | Article |
| id | doaj-art-0be81f54705241a1b2eb93aee8df4201 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-0be81f54705241a1b2eb93aee8df42012025-02-03T06:06:56ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/618968618968Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet SpaceA. Abdollahi0S. Mehrangiz1Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, IranDepartment of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, IranLet 𝜑(𝑧)=𝑧𝑛,𝑧∈𝕌, for some positive integer 𝑛 and 𝐶𝜑 the composition operator on the Dirichlet space 𝒟 induced by 𝜑. In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators 𝐶∗𝜑𝐶𝜑,𝐶𝜑𝐶∗𝜑 and self-commutators of 𝐶𝜑, which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.http://dx.doi.org/10.1155/2011/618968 |
| spellingShingle | A. Abdollahi S. Mehrangiz Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space Abstract and Applied Analysis |
| title | Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space |
| title_full | Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space |
| title_fullStr | Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space |
| title_full_unstemmed | Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space |
| title_short | Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space |
| title_sort | self commutators of composition operators with monomial symbols on the dirichlet space |
| url | http://dx.doi.org/10.1155/2011/618968 |
| work_keys_str_mv | AT aabdollahi selfcommutatorsofcompositionoperatorswithmonomialsymbolsonthedirichletspace AT smehrangiz selfcommutatorsofcompositionoperatorswithmonomialsymbolsonthedirichletspace |