Duality in some vector-valued function spaces
As usual, \(D\) denotes the unit disk, and for a complex Banach space \(X\), \(H^ p(D,X)\) is the \(X\)-valued Hardy space. In Section 2 of the article, the author shows that \(H^ p(D,X^*)\) is always (isometrically isomorphic to) a dual space for arbitrary \(X\) and \(1\leq p\leq\infty\). He uses this to characterize the analytic Radon-Nikodým property (aRNP) in dual Banach spaces. [For a thorough discussion of vector valued Hardy spaces and the aRNP, see \textit{W. Hensgen}, Hardy-Räume vektorwertiger Funktionen, Dissertation, Univ. München (1986; Zbl 0594.46028).] In Section 3, the author turns to the \(\Lambda\)-Radon-Nikodým property introduced by G. A. Edgar; here \(G\) is a compact metrizable abelian group and \(\Lambda\) is a subset of the dual group \(\Gamma\) of \(G\). Let \(L_ \Lambda^ p(G,X)\) be the space of those functions \(f\in L^ p(G,X)\) for which the Fourier coefficients \(\widehat f(\gamma)\) vanish for all \(\gamma\not\in\Lambda\). If \(\Lambda\) is a Sidon set in \(\Gamma\), every Banach space not containing a copy of \(c_ 0\) has the \(\Lambda\)-Radon- Nikodým property. For an infinite Sidon subset \(\Lambda\subset\Gamma\), an arbitrary Banach space \(X\) and \(1\leq p\leq\infty\), it is proved that \(L_ \Lambda^ p(G,X^*)\) is a dual space if and only if \(X^*\) does not contain a copy of \(c_ 0\).
Banach spaces not containing a copy of \(c_ 0\)
scientific article; zbMATH DE number 97422
scientific article