×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] DOI: 10.1090/S0002-9939-2010-10508-7 · Zbl 1270.47018 · doi:10.1090/S0002-9939-2010-10508-7
[2] DOI: 10.2307/1970850 · Zbl 0253.46049 · doi:10.2307/1970850
[3] DOI: 10.1090/S0002-9947-1940-0004094-3 · JFM 66.0554.01 · doi:10.1090/S0002-9947-1940-0004094-3
[4] Lindenstruss, Studia Math. 29 pp 275– (1968)
[5] Kwapień, Studia Math. 29 pp 327– (1968)
[6] DOI: 10.1090/conm/002/621850 · doi:10.1090/conm/002/621850
[7] Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. (1955)
[8] DOI: 10.4064/sm197-3-6 · Zbl 1190.47024 · doi:10.4064/sm197-3-6
[9] DOI: 10.4153/CJM-1953-017-4 · Zbl 0050.10902 · doi:10.4153/CJM-1953-017-4
[10] DOI: 10.1016/j.jmaa.2008.12.047 · Zbl 1168.46008 · doi:10.1016/j.jmaa.2008.12.047
[11] Gelfand, Rev. Roumaine Math. Pures Appl. 5 pp 742– (1938)
[12] DOI: 10.1090/S0002-9947-1940-0002020-4 · JFM 66.0556.01 · doi:10.1090/S0002-9947-1940-0002020-4
[13] DOI: 10.1006/jmaa.1994.1246 · Zbl 0878.46009 · doi:10.1006/jmaa.1994.1246
[14] DOI: 10.1017/CBO9780511526138 · doi:10.1017/CBO9780511526138
[15] Castillo, Rev. Mat. Univ. Comput. Madrid 6 pp 43– (1993)
[16] DOI: 10.1002/mana.19841190105 · Zbl 0601.47019 · doi:10.1002/mana.19841190105
[17] DOI: 10.4153/CJM-1955-032-1 · Zbl 0068.09301 · doi:10.4153/CJM-1955-032-1
[18] DOI: 10.4064/sm150-1-3 · Zbl 1008.46008 · doi:10.4064/sm150-1-3
[19] Saphar, Studia Math. 38 pp 71– (1970)
[20] Pietsch, Studia Math. 28 pp 333– (1967)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.