Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras
Let 𝑋 and 𝐾 be compact plane sets with 𝐾⊆𝑋. We define 𝐴(𝑋,𝐾)={𝑓∈𝐶(𝑋)∶𝑓|𝐾∈𝐴(𝐾)}, where 𝐴(𝐾)={𝑔∈𝐶(𝑋)∶𝑔 is analytic on int(𝐾)}. For 𝛼∈(0,1], we define Lip(𝑋,𝐾,𝛼)={𝑓∈𝐶(𝑋)∶𝑝𝛼,𝐾(𝑓)=sup{|𝑓(𝑧)−𝑓(𝑤)|/|𝑧−𝑤|𝛼∶𝑧,𝑤∈𝐾,𝑧≠𝑤}<∞} and Lip𝐴(𝑋,𝐾,𝛼)=𝐴(𝑋,𝐾)∩Lip(𝑋,𝐾,𝛼). It is known that Lip𝐴(𝑋,𝐾,𝛼) is a natural Banach f...
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2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/146758 |
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| author | Davood Alimohammadi Maliheh Mayghani |
| author_facet | Davood Alimohammadi Maliheh Mayghani |
| author_sort | Davood Alimohammadi |
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| description | Let 𝑋 and 𝐾 be compact plane sets with 𝐾⊆𝑋. We define 𝐴(𝑋,𝐾)={𝑓∈𝐶(𝑋)∶𝑓|𝐾∈𝐴(𝐾)}, where 𝐴(𝐾)={𝑔∈𝐶(𝑋)∶𝑔 is analytic on int(𝐾)}. For 𝛼∈(0,1], we define Lip(𝑋,𝐾,𝛼)={𝑓∈𝐶(𝑋)∶𝑝𝛼,𝐾(𝑓)=sup{|𝑓(𝑧)−𝑓(𝑤)|/|𝑧−𝑤|𝛼∶𝑧,𝑤∈𝐾,𝑧≠𝑤}<∞} and Lip𝐴(𝑋,𝐾,𝛼)=𝐴(𝑋,𝐾)∩Lip(𝑋,𝐾,𝛼). It is known that Lip𝐴(𝑋,𝐾,𝛼) is a natural Banach function algebra on 𝑋 under the norm ||𝑓||Lip(𝑋,𝐾,𝛼)=||𝑓||𝑋+𝑝𝛼,𝐾(𝑓), where ||𝑓||𝑋=sup{|𝑓(𝑥)|∶𝑥∈𝑋}. These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of Lip𝐴(𝑋1,𝐾1,𝛼1) to natural Banach function subalgebras of Lip𝐴(𝑋2,𝐾2,𝛼2) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of Lip𝐴(𝑋,𝐾,𝛼). |
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| spelling | doaj-art-5332c0e33db24ca19460003e088ecbb32025-02-03T05:45:10ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/146758146758Unital Compact Homomorphisms between Extended Analytic Lipschitz AlgebrasDavood Alimohammadi0Maliheh Mayghani1Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, IranDepartment of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, IranLet 𝑋 and 𝐾 be compact plane sets with 𝐾⊆𝑋. We define 𝐴(𝑋,𝐾)={𝑓∈𝐶(𝑋)∶𝑓|𝐾∈𝐴(𝐾)}, where 𝐴(𝐾)={𝑔∈𝐶(𝑋)∶𝑔 is analytic on int(𝐾)}. For 𝛼∈(0,1], we define Lip(𝑋,𝐾,𝛼)={𝑓∈𝐶(𝑋)∶𝑝𝛼,𝐾(𝑓)=sup{|𝑓(𝑧)−𝑓(𝑤)|/|𝑧−𝑤|𝛼∶𝑧,𝑤∈𝐾,𝑧≠𝑤}<∞} and Lip𝐴(𝑋,𝐾,𝛼)=𝐴(𝑋,𝐾)∩Lip(𝑋,𝐾,𝛼). It is known that Lip𝐴(𝑋,𝐾,𝛼) is a natural Banach function algebra on 𝑋 under the norm ||𝑓||Lip(𝑋,𝐾,𝛼)=||𝑓||𝑋+𝑝𝛼,𝐾(𝑓), where ||𝑓||𝑋=sup{|𝑓(𝑥)|∶𝑥∈𝑋}. These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of Lip𝐴(𝑋1,𝐾1,𝛼1) to natural Banach function subalgebras of Lip𝐴(𝑋2,𝐾2,𝛼2) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of Lip𝐴(𝑋,𝐾,𝛼).http://dx.doi.org/10.1155/2011/146758 |
| spellingShingle | Davood Alimohammadi Maliheh Mayghani Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras Abstract and Applied Analysis |
| title | Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras |
| title_full | Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras |
| title_fullStr | Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras |
| title_full_unstemmed | Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras |
| title_short | Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras |
| title_sort | unital compact homomorphisms between extended analytic lipschitz algebras |
| url | http://dx.doi.org/10.1155/2011/146758 |
| work_keys_str_mv | AT davoodalimohammadi unitalcompacthomomorphismsbetweenextendedanalyticlipschitzalgebras AT malihehmayghani unitalcompacthomomorphismsbetweenextendedanalyticlipschitzalgebras |