Extremal problems for the vector-valued \(\langle L^ 1/H_ 0^ 1,H^ \infty\rangle\) duality (Q1908220)

Let \(B\) be a Banach space. A closed subset \(A\) of \(B\) is said to be (i) proximinal, (ii) semi-Chebyshev, (iii) Chebyshev if for every \(x\in B\) there exists (i) at least one, (ii) at most one, (iii) exactly one best approximation in \(A\). A classical theorem of \textit{J. L. Doob} [Duke Math. J. 8,413-424 (1941; Zbl 0063.01144)] states that the Hardy space \[ H^1_0(\mathbb{T})= \{f\in L^1(\mathbb{T}): \widehat f(n)= 0\quad \forall n\leq 0\} \] is a Chebyshev subspace of \(L^1(\mathbb{T})\). One of the aims of the paper under review is to give vector valued generalizations of this theorem. To this end, let \(X\) be a complex Banach space and let \(L^1(X)= L^1(\mathbb{T}, X)\) be the Bochner space of \(X\)-valued functions on the unit circle \(\mathbb{T}\). Associated is its Hardy space \(H^1_0(X)\) defined analogously as above. The main result is now the following: Let \(X\) be a complex Banach space having the analytic Radon-Nikodym property (ARNP). Assume that \(X\) is contractively complemented in its bidual \(X^{**}\). Then \(H^1_0(X)\) is a proximinal subspace of \(L^1(X)\). On the other hand, if \(X\) does not have the ARNP then the conclusion may fail. In fact the author shows that for \(X= L^1(\mathbb{T})/H^1_0(\mathbb{T})\), a space which does not have the ARNP [see the author's thesis ``Hardy-Räume vektorwertiger Funktionen'', Univ. München (1986; Zbl 0594.46028)], the associate Hardy space \(H^1_0(X)\) is not proximinal. It is also shown that \(H^1_0(X)\) is semi-Chebyshev if \(X\) is strictly convex. Other results on extremal functions are included.