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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 $$\mu$$.

##### MSC:
 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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