Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras - Davood Alimohammadi; Maliheh Mayghani (Abstract and Applied Analysis)
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).