Pełczyński's property $V$ for spaces of compact operators
A bounded linear operator \(T : X \to Y\) between two Banach spaces is said to be unconditionally converging if for every unconditional Cauchy series \(\sum_n x_n\) in \(X\), the series \(\sum_n T x_n\) is unconditionally convergent. A Banach space \(X\) has Pełczyński's property (\(V\)) if for all Banach spaces \(Y\), every unconditionally converging operator \(T : X \to Y\) is weakly compact. In this paper the authors prove that if \(X\) is a reflexive Banach space with an unconditional basis, then the Banach space of compact operators on \(X\), \(K(X)\), has Pełczyński's property (\(V\)). In particular, \(K(L_p[0,1])\), for \(1 < p < \infty\), has Pełczyński's property (V). The proof uses a sequential representation of the space of compact operators on \(X\).
scientific article; zbMATH DE number 7382090