Unital compact homomorphisms between extended analytic Lipschitz algebras (Q642684)

Summary: Let \(X\) and \(K\) be compact plane sets with \(K \subseteq X\). We define \(A(X, K) = \{f \in C(X) : f|_K \in A(K)\}\), where \(A(K) = \{g \in C(K) : g\) is analytic on \(\text{int}(K)\}\). For \(\alpha \in (0, 1]\), we define \(\text{Lip}(X, K, \alpha) = \{f \in C(X) : p_{\alpha,K}(f) = \text{sup}\{|f(z) - f(w)|/|z - w|^{\alpha} : z, w \in K, z \neq w\} < \infty \}\) and \(\text{Lip}_A(X, K, \alpha) = A(X, K) \cap \text{Lip}(X, K, \alpha)\). It is known that \(\text{Lip}_A(X, K, \alpha)\) is a natural Banach function algebra on \(X\) under the norm \(||f||_{\text{Lip}(X, K, \alpha)} = ||f||_X + p_{\alpha, K}(f)\), where \(||f||_X = \text{sup}\{|f(x)| : x \in X\}\). These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of \(\text{Lip}_A(X_1, K_1, \alpha_1)\) to natural Banach function subalgebras of \(\text{Lip}_A(X_2, K_2, \alpha_2)\) and investigate necessary and sufficient conditions under which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of \(\text{Lip}_A(X, K, \alpha)\).