\(p\)-converging operators and Dunford-Pettis property of order \(p\)
Let \(X\) be a Banach space and, for \(1\leq p <\infty\), denote by \(\ell_p^{w}(X)\) the Banach space of all weakly \(p\)-summable sequences in \(X\). \textit{J. M. F. Castillo} and \textit{F. Sánchez} [Rev. Mat., 6, 43--59 (1993; Zbl 0807.46033)] introduced the ideal \(\mathcal C_p \) of \(p\)-converging operators as those which transform weakly \(p\)-summable sequences into norm null sequences. These ideals lie between the ideal \(\mathcal C_1 \) of unconditionally converging operators and that of \(\mathcal C _{\infty}\), the ideal of completely continuous operators. Also, Castillo and Sánchez introduced the notion of Dunford-Pettis property of order~\(p\). A~Banach space \(X\) has the Dunford-Pettis property of order~\(p\) if, for every Banach space \(Y\), every weakly compact operator \(T : X \to Y\) belongs to \(\mathcal C_p (X; Y)\). This notion generalizes the classical Dunford-Pettis property, which, following this notation, is the Dunford-Pettis property of order~\(\infty \). In the present article, among other things, the authors introduce the notions of weakly \(p\)-Cauchy sequence (a sequence \((x_n)_n \subset X\) is weakly \(p\)-Cauchy if, for each pair of strictly increasing sequences \((k_n)_n\) and \((j_n)_n \) of positive integers, the sequence \( (x_ {k_n} -x_ {j_n})_ n \in \ell_p^{w} (X) \)) and of conditionally weakly \( p\)-compact set (a set \( K \subset X \) is conditionally \(p\)-compact if every sequence in \(K\) has a weakly \(p\)-Cauchy subsequence). They use these notions to characterize \(p\)-converging operators and Banach spaces with the Dunford-Pettis property of order \(p\). Also, the hereditary Dunford-Pettis property of order \(p\), the Banach-Saks property of order \(p\), and the \(p$-$(V)\) sets are considered and the authors obtain \(p\)-analogous results of well-known results on this topic. Finally, as it was done in [\textit{M. Kačena} et al., Adv. Math., 234, 488--527 (2013; Zbl 1266.46007)] for the Dunford-Pettis property, the authors introduce a locally convex topology in order to quantify the Dunford-Pettis property of order~\(p\).
hereditary Dunford-Pettis property of order \(p\)
scientific article; zbMATH DE number 6852146
scientific article