Rank \(\alpha\) operators on the space \(C(T,X)\) (Q2773348)

The author is interested in the study of rank \(\alpha\) operators, \(0\leq\alpha <1 \), between Banach spaces which are defined as follows. Let \(X\) be a Banach space and \(0\leq\alpha<1\). Recall that a sequence \((x_n)_{n\in\mathbb N}\subset X\) is called \(\tau_\alpha\)-convergent to 0 if there exists a constant \(c>0\) such that \(\|\sum_{n\in B}x_n\|\leq c| B|^\alpha\), for all finite subsets \(B \subset\mathbb N\); a sequence \((x_n)_{n\in\mathbb N}\subset X\) is called \(\tau_\alpha \)-convergent to \(x\) if \((x_n-x)_{n\in\mathbb N}\) is \(\tau_\alpha\)-convergent to 0. The notion of \(\tau_\alpha\)-convergence has been first introduced by \textit{A. Pełczyńsky} [Stud. Math. 16, 173--182 (1957; Zbl 0080.09701)]. For \(0 \leq\alpha<1\), an operator \(U\in L(X,Y)\) is called a rank \(\alpha\) operator if for every sequence \((x_n)_{n\in\mathbb N} \subset X\) \(\tau_\alpha\)-convergent to \(x\), \((Ux_n)_{n\in\mathbb N}\) converges to \(Ux\) in norm.NEWLINENEWLINE In this paper, the author collects some results on rank \(\alpha\) operators, \(0\leq\alpha<1\), including an interpolation result and a characterization of rank \(\alpha\) operators \(U:C(T,X) \to Y\) in terms of their representing measures, where \(T\) is a compact Hausdorff space, \(X,Y\) are Banach spaces and \(C(T,X)\) is the Banach space of all \(X\)-valued continuous functions defined on \(T\), endowed with the supremum norm.