Compact composition operators on $B(D).$

Abstract:

Let $D$ be a domain in the complex plane, $\phi :D \to D$ be analytic, and $B(D)$ be the uniform algebra of bounded analytic functions on $D$ with maximal ideal space $M$. The composition operator ${C_\phi }(f) = f \circ \phi$ is compact if and only if the weak* and norm closures of $\phi (D)$ coincide if and only if whenever the Euclidean closure of $\phi (D)$ contains a point $\lambda$ of the boundary of $D$ then each $f \in B(D)$ extends continuously from $\phi (D)$ to $\lambda$. If ${C_\phi }$ is compact, then either $\phi$ fixes a point of $D$ or else the adjoint of ${C_\phi }$ fixes a point of $M$.