Composition Operator on Bergman-Orlicz Space
Let 𝔻 denote the open unit disk in the complex plane and let dA(z) denote the normalized area measure on 𝔻. For α>−1 and Φ a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on [0,∞), the...
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Series: | Journal of Inequalities and Applications |
Online Access: | http://dx.doi.org/10.1155/2009/832686 |
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doaj-3efe586b877540a096e2c7a3de37fb6f2020-11-25T00:18:54ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2009-01-01200910.1155/2009/832686Composition Operator on Bergman-Orlicz SpaceGuangfu CaoZhijie JiangLet 𝔻 denote the open unit disk in the complex plane and let dA(z) denote the normalized area measure on 𝔻. For α>−1 and Φ a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on [0,∞), the Bergman-Orlicz space LαΦ is defined as follows LαΦ={f∈H(𝔻):∫𝔻Φ(log+|f(z)|)(1−|z|2)αdA(z)<∞}. Let φ be an analytic self-map of 𝔻. The composition operator Cφ induced by φ is defined by Cφf=f∘φ for f analytic in 𝔻. We prove that the composition operator Cφ is compact on LαΦ if and only if Cφ is compact on Aα2, and Cφ has closed range on LαΦ if and only if Cφ has closed range on Aα2. http://dx.doi.org/10.1155/2009/832686 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Guangfu Cao Zhijie Jiang |
spellingShingle |
Guangfu Cao Zhijie Jiang Composition Operator on Bergman-Orlicz Space Journal of Inequalities and Applications |
author_facet |
Guangfu Cao Zhijie Jiang |
author_sort |
Guangfu Cao |
title |
Composition Operator on Bergman-Orlicz Space |
title_short |
Composition Operator on Bergman-Orlicz Space |
title_full |
Composition Operator on Bergman-Orlicz Space |
title_fullStr |
Composition Operator on Bergman-Orlicz Space |
title_full_unstemmed |
Composition Operator on Bergman-Orlicz Space |
title_sort |
composition operator on bergman-orlicz space |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1025-5834 1029-242X |
publishDate |
2009-01-01 |
description |
Let 𝔻 denote the open unit disk in the complex plane and let dA(z) denote the normalized area measure on 𝔻. For α>−1 and Φ a twice differentiable, nonconstant, nondecreasing, nonnegative, and convex function on [0,∞), the Bergman-Orlicz space LαΦ is defined as follows LαΦ={f∈H(𝔻):∫𝔻Φ(log+|f(z)|)(1−|z|2)αdA(z)<∞}. Let φ be an analytic self-map of 𝔻. The composition operator Cφ induced by φ is defined by Cφf=f∘φ for f analytic in 𝔻. We prove that the composition operator Cφ is compact on LαΦ if and only if Cφ is compact on Aα2, and Cφ has closed range on LαΦ if and only if Cφ has closed range on Aα2. |
url |
http://dx.doi.org/10.1155/2009/832686 |
work_keys_str_mv |
AT guangfucao compositionoperatoronbergmanorliczspace AT zhijiejiang compositionoperatoronbergmanorliczspace |
_version_ |
1725374445841809408 |