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An operator summability of sequences in Banach spaces. (English) Zbl 1301.46004
Let \(X\) and \(Y\) be Banach spaces, \(1 \leq p < \infty\), and \(p'\) the conjugate of \(p\), i.e., \(\frac{1}{p} + \frac{1}{p'} = 1\). Denote by \(B(X,Y)\) the space of bounded linear operators from \(X\) to \(Y\).
Recall two well-known summability properties: A sequence \((x_n)\) in \(X\) is said to be norm \(p\)-summable if \(\sum_{n=1}^\infty \|x_n\|^p < \infty\) and \((x_n)\) is said to be weakly \(p\)-summable if, for every \(f \in X^*\), we have \(\sum_{n=1}^\infty |f(x_n)|^p < \infty\). Clearly, norm \(p\)-summable sequences are weakly \(p\)-summable, and it is also a fact that every weakly \(p\)-summable sequence in \(X\) is norm \(p\)-summable iff \(X\) is of finite dimension.
The authors introduce an intermediate summability property between the norm \(p\)-summability and the weak \(p\)-summability properties: a sequence \((x_n)\) in \(X\) is operator \(p\)-summable if, for every \(T \in B(X, \ell_p)\), we have \(\sum_{n=1}^\infty \|Tx_n\|^p < \infty\).
It is proved that every weakly \(p\)-summable sequence in \(X\) is operator \(p\)-summable (i.e., \(X\) is a weak \(p\)-space) iff \(\Pi_p(X,\ell_p) = B(X, \ell_p)\). (Here, \(\Pi_p(X,\ell_p)\) denotes the subset of \(B(X,\ell_p)\) consisting of the absolutely \(p\)-summing operators.) A list of other characterizations of weak \(p\)-spaces is provided as well. One of these characterizations is in terms of a \(p\)-version of the Dunford-Pettis property (\(p\)-DPP) which the authors introduce. As an \(\mathcal L_\infty\) space, \(X\) has the \(p\)-DPP for \(1 \leq p \leq 2\), and the authors are able to provide examples of weak \(p\)-spaces.
Furthermore it is proved that every operator \(p\)-summable sequence is norm \(p\)-summable iff \(\Pi_p^d(\ell_{p'}, X) = \Pi_p(\ell_{p'},X)\). (Here, \(\Pi_p^d(\ell_{p'},X)\) denotes the subset of \(B(\ell_{p'}, X)\) consisting of the operators whose adjoints are absolutely \(p\)-summing.) In turn, it is known from A. K. Karn and D. P. Sinha [Glasg. Math. J. 56, No. 2, 427–437 (2014; Zbl 1301.46004)] that \(\Pi_p^d(\ell_{p'}, X) = \Pi_p(\ell_{p'}, X)\) precisely when \(X\) is a subspace of \(L_p(\mu)\) for some Borel measure \(\mu\).

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B50 Compactness in Banach (or normed) spaces
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