Extremal Problems for the Vector-Valued 〈L1/H01, H∞〉 Duality

LetXbe a complex Banach space andL1(X)≔L1(T; X) the Bochner space on the circle T. TheX-valued Hardy space H01(X)≔[f∈L1(X): f(n)=0 ∀n⩽0] is proximinal inL1(X) ifHhas ARNP and is contractively complemented inX″. It is semi-Chebyshev ifXis strictly convex. WithH∞(X′) the dual space ofL1(X)/H01(X), extremal kernels and functions for this duality are studied. Proximinality fails forX≔L1/H01; this is equivalent to the assertion that forΛ≔N×Z∪Z×N,LΛ1(T2) is not proximinal inL1(T2). A class of subsetsΛ⊂Z2is described for which this non-proximinality holds.