Unital Compact Homomorphisms between Extended Analytic Lipschitz Algebras

Abstract EN

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).