Let Δ be the open unit disc in
C, ∂Δ its boundary and B ⊂ ∂Δ a relatively open set. Let X be a complex Banach
space. Denote by HB(Δ,X) the set of all continuous functions from Δ ∪ B to X
which are analytic on Δ. A set P ⊂ X is said to have the analytic extension property
with respect to HB(Δ,X) if for each relatively closed set F ⊂ B of Lebesgue measure
0 and for each continuous function f : F → P there exists g ∈ HB(Δ,X) with
g∣F = f and g(Δ ∪ B) ⊂ P.
Theorem. Let P ⊂ X be an open set. Then P has the analytic extension property
with respect to HB(Δ,X) for every relatively open B ⊂ ∂Δ if and only if P is
connected.
