Dual spaces of compact operator spaces and the weak Radon-Nikodym property (Q2919698)

Let \(X\) and \(Y\) be Banach spaces and \(\mathcal K(X,Y)\) the space of compact operators from \(X\) to \(Y\).NEWLINENEWLINERecall that the dual \(X^{\ast}\) of \(X\) is said to have the (weak) Radon-Nikodým property if, for each complete finite measure space \((\Omega, \Sigma, \mu)\) and each \(\mu\)-continuous \(X^{\ast}\)-valued countably additive vector measure \(\nu :\Sigma \to X^{\ast}\) of bounded variation, there exists a (Pettis-) integrable function \(f :\Omega \to X^{\ast}\) such that \(\nu(E) = (P-)\int_E f\, \text{d}\mu\) for all \(E \in \Sigma\).NEWLINENEWLINEThe following theorem of Feder and Saphar, which describes the dual of \(\mathcal K(X,Y)\) when \(X^{\ast\ast}\) or \(Y^{\ast}\) has the Radon-Nikodým property, is well known: If \(X^{\ast\ast}\) or \(Y^\ast\) has the Radon-Nikodým property, then for every \(\phi \in K(X,Y)^\ast\) and \(\varepsilon > 0\), there are sequences \((x_n^{\ast\ast}) \subset X^{\ast\ast}\) and \((y_n^{\ast}) \subset Y^{\ast}\) such that \(\phi(T) = \sum_{n=1}^\infty x_n^{\ast\ast}T^{\ast}(y_n^{\ast})\) for all \(T \in \mathcal K(X,Y)\) and \(\sum_{n=1}^\infty \|x_n^{\ast\ast}\|\|y_n^{\ast}\| < \|\phi\| + \varepsilon\).NEWLINENEWLINEIn this paper, a similar result (Theorem~3.5) is proved for the case when \(X^{\ast\ast}\) or \(Y^{\ast}\) has the weak Radon-Nikodým property: If \(X^{\ast\ast}\) or \(Y^\ast\) has the weak Radon-Nikodým property, then for every \(\phi \in K(X,Y)^\ast\), there are sequences of sequences \(((x_{i, n}^{\ast\ast})_{i=1}^{m_n})_{n=1}^\infty\) and \(((y_{i,n}^{\ast})_{i=1}^{m_n})_{n=1}^\infty\), with \(x_{i, n}^{\ast\ast} \in X^{\ast\ast}\) and \(y_{i,n}^{\ast} \in Y^{\ast}\), such that \(\phi(T) = \lim_{n \to \infty}\sum_{i=1}^{m_n} x_{i,n}^{\ast\ast}T^{\ast}(y_{i,n}^{\ast})\) for all \(T \in \mathcal K(X,Y)\) and \(\limsup_{n \to \infty}\sum_{i=1}^{m_n} \|x_{i,n}^{\ast\ast}\|\|y_{i,n}^{\ast}\| \leq \|\phi\|\).NEWLINENEWLINEThis result is then used to obtain, among other things, a similar representation of the dual of \(\mathcal K_{\omega^{\ast}\omega}(X^{\ast}, Y)\), the weak\(^\ast\)-weak continuous compact operators from \(X^{\ast}\) to \(Y\), when \(X^{\ast\ast}\) or \(Y^{\ast}\) has the weak Radon-Nikodým property.