Geometric properties of \(h^p(\mathbb D, X)\) and generalizations
The author first observes that the Hardy space of \(L^1(\mathbb T)\)-valued harmonic functions \(h^2(\mathbb D, L^1(\mathbb T))\) admits an equivalent locally uniformly convex norm, but \(L^1(\mathbb T)\) fails the RNP. This answers in the negative a question posed by \textit{P. N. Dowling} [\textit{Z.-B. Hu} and \textit{M. A. Smith}, Can. J. Math. 46, No.~2, 274--283 (1994; Zbl 0802.46024)]. The author then proposes the question whether the fact that \(H^p(\mathbb D, X)\) has an equivalent LUR norm implies the ARNP of \(X\). This holds for Banach lattices \(X\). It is shown that if \(X^*\) has the ARNP, then \(h^p(\mathbb D, X^*)\) has the ARNP whenever \(Y^*\) has some extra property. Several modifications of the Kadec-Klee property and the Radon-Nikodým property on sets are studied for the space \(h^p(\mathbb D, X)\) (or its formulation in terms of cone-absolutely summing operators from \(L^{p'}(\mathbb T)\) to \(X\), denoted by \(VB^p(\mathbb T,X))\). With this formulation in terms of operators, the author considers the more general setting of compact abelian groups \(G\) and studies such properties for their subspaces \(VB_{\Lambda}^p(\mathbb T,X)\), where the operators are assumed to satisfy \(T(\bar \lambda)=0\) for \(\lambda \notin \Lambda\), and for quotients \(VB_{\Lambda_2}^p(\mathbb T,X)/VB_{\Lambda_1}^p(\mathbb T,X) \) of such spaces where \(\Lambda_1\subset \Lambda_2\).
scientific article; zbMATH DE number 5931129
scientific article