Fredholm weighted composition operators on analytic Lipschitz algebras

Abstract

In this paper, we will first show that any weighted composition operator on analytic Lipschitz algebras and on analytic little Lipschitz algebras is injective provided the underlying compact set has dense interior, and we characterize completely the surjectivity of them. Then we will give necessary conditions, and in a special case, a sufficient condition for a weighted composition operator on these algebras to be Fredholm. Finally, we note that all the results obtained about analytic (little) Lipschitz algebras are also valid for the uniform algebra $A\left(X\right)$ where X is a compact subset of the complex plane $C$.