Pointwise absolutely convergent series of operators and related classes of Banach spaces (Q424097)

Let \(G_0(X,Y)\) denote the set of all bounded linear operators from a Banach space \(X\) to a Banach space \(Y\) of rank~\(1\). Define inductively \(G_n(X,Y)\) to be the space of those operators having a representation \(Tx=\sum_{k=1}^\infty T_k x\) with \(T_k\in G_{n-1}(X,Y)\) such that all the series \(\sum_k T_k x\) are absolutely convergent. The paper studies these classes of operators and the classes \({\mathcal G}_n\) of Banach spaces \(X\) for which the identity operator belongs to \(G_n(X,X)\); equivalently \(G_n(X,X)=L(X)\). The main results of the paper are the following: (1) If \(\dim X =\infty\), then \(X\) belongs to \({\mathcal G}_1\) if and only if \(X\) is isomorphic to \(\ell_1\). (2) If \(T\in G_n(X,Y)\) for some \(n\), then \(T\) is a Dunford-Pettis operator, i.e., \(T\) maps weakly null sequences to norm null sequences. Consequently, if \(X\in {\mathcal G}_n\) for some \(n\), then \(X\) has the Schur property. (3) If \(X\in {\mathcal G}_n\) for some \(n\), then \(X\) has the approximation property. If \(X\) is reflexive with the approximation property, then \(G_2(X,X)= G_3(X,X)= \dots\). (4) For each \(n\), there is a Banach space \(X_n\in {\mathcal G}_n \setminus {\mathcal G}_{n-1}\). Hence all the inclusions \({\mathcal G}_1 \subset {\mathcal G}_2 \subset \dots\) are strict. The paper complements investigations in [\textit{V.~Kadets}, \textit{N.~Kalton} and \textit{D.~Werner}, Bull.\ Lond.\ Math.\ Soc.\ 37, No.~2, 265--274 (2005; Zbl 1077.46008)], where pointwise unconditionally convergent series of operators were studied.