Some Banach space properties of the duals of the disk algebra and \(H^\infty\)
Following Pełczyński, a Banach space \(X\) is said to have property (V) if every subset \(W\) of the dual \(X^\ast\) satisfying \[ \lim_{n}\sup_{x^\ast\in W}|\langle x^\ast,x_n\rangle|=0 \] for every weakly unconditionally Cauchy series \(\sum_nx_n\) in \(X\) is weakly relatively compact in \(X^\ast\). A Banach space \(X\) has the local property (V) if each ultrapower \((X)_{\mathcal U}\) has the property (V). A Banach space \(X\) is said to be a Grothendieck space if every sequence in \(X^\ast\) that is weak-\(\ast\) convergent is also weakly convergent. The authors proved that the disk algebra \(A\) has the local property (V). From this fact it follows that \(H^\infty\) also has the local property (V). This result is an extension of a \textit{J. Bourgain}'s theorem [Stud. Math. 75, 193-216 (1983; Zbl 0533.46035)]. It is also proved that the odd duals of \(A\) and \(H^\infty\) are Grothendieck spaces.
scientific article; zbMATH DE number 1548304