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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 146758, 15 pages
http://dx.doi.org/10.1155/2011/146758
Research Article

Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 22 May 2011; Accepted 10 August 2011

Academic Editor: Malisa R. Zizovic

Copyright © 2011 Davood Alimohammadi and Maliheh Mayghani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑋 and 𝐾 be compact plane sets with 𝐾 βŠ† 𝑋 . We define 𝐴 ( 𝑋 , 𝐾 ) = { 𝑓 ∈ 𝐢 ( 𝑋 ) ∢ 𝑓 | 𝐾 ∈ 𝐴 ( 𝐾 ) } , where 𝐴 ( 𝐾 ) = { 𝑔 ∈ 𝐢 ( 𝑋 ) ∢ 𝑔 is analytic on i n t ( 𝐾 ) } . For 𝛼 ∈ ( 0 , 1 ] , we define L i p ( 𝑋 , 𝐾 , 𝛼 ) = { 𝑓 ∈ 𝐢 ( 𝑋 ) ∢ 𝑝 𝛼 , 𝐾 ( 𝑓 ) = s u p { | 𝑓 ( 𝑧 ) βˆ’ 𝑓 ( 𝑀 ) | / | 𝑧 βˆ’ 𝑀 | 𝛼 ∢ 𝑧 , 𝑀 ∈ 𝐾 , 𝑧 β‰  𝑀 } < ∞ } and L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) = 𝐴 ( 𝑋 , 𝐾 ) ∩ L i p ( 𝑋 , 𝐾 , 𝛼 ) . It is known that L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) is a natural Banach function algebra on 𝑋 under the norm | | 𝑓 | | L i p ( 𝑋 , 𝐾 , 𝛼 ) = | | 𝑓 | | 𝑋 + 𝑝 𝛼 , 𝐾 ( 𝑓 ) , where | | 𝑓 | | 𝑋 = s u p { | 𝑓 ( π‘₯ ) | ∢ π‘₯ ∈ 𝑋 } . These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of L i p 𝐴 ( 𝑋 1 , 𝐾 1 , 𝛼 1 ) to natural Banach function subalgebras of L i p 𝐴 ( 𝑋 2 , 𝐾 2 , 𝛼 2 ) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) .