Analytic extensions and selections

Abstract:

Let G be a closed subset of the closed unit disc in C, let F be a closed subset of the unit circle of measure 0 and let $\Phi$ map G into the class of all open subsets of a complex Banach space X. Under suitable additional assumptions on $\Phi$ we prove that given any continuous function $f: F \to X$ satisfying $f(z) \in {\text {closure(}}\Phi (z)) (z \in F \cap G)$ there exists a continuous function f from the closed unit disc into X, analytic in the open unit disc, which extends f and satisfies $\tilde f(z) \in \Phi (z) (z \in G - F)$. This enables us to generalize and sharpen known dominated extension theorems for the disc algebra.