Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Let and be compact plane sets with ⊆. We define (,)={∈()∶|∈()}, where ()={∈()∶ is analytic on int()}. For ∈(0,1], we define Lip(,,)={∈()∶,()=sup{|()−()|/|−|∶,∈,≠}<∞} and Lip(,,)=(,)∩Lip(,,). It is known that Lip(,,) is a natural Banach function algebra on under the norm ||||Lip(,,)=||||+,(), where ||||=sup{|()|∶∈}. These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of Lip(1,1,1) to natural Banach function subalgebras of Lip(2,2,2) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of Lip(,,)