zbMATH — the first resource for mathematics

Cohen and multiple Cohen strongly summing multilinear operators. (English) Zbl 1315.47078
The operator ideal of absolutely \(p\)-summing linear operators is not symmetric or, in other words, not closed under conjugation. In [Math. Ann. 201, 177–200 (1973; Zbl 0233.47019)], J. S. Cohen introduced the notion of so-called Cohen strongly \(p\)-summing sequences, as sequences \((x_i)\) in a Banach space \(E\) such that the series \(\sum_{i=1}^\infty \varphi _i(x_i)\) converges for all \((\varphi _i) \in \ell _{p^\ast }^w(E')\) with \(1/p+1/{p^\ast }=1\). The space \(\ell _p\langle E \rangle \) of all Cohen strongly \(p\)-summing sequences in \(E\) is equipped with a suitable norm. For \(1<p<\infty \), an operator \(T\in {\mathcal L}(E,F)\) is called Cohen strongly \(p\)-summing if \((Tx_i) \in \ell _p\langle F\rangle \) for all \((x_i) \in \ell _p(E)\). It is shown that an operator \(T\in {\mathcal L}(E,F)\) is absolutely \(p\)-summing if and only if \(T'\) is strongly \(p\)-summing, \(1\leq p < \infty \).
In this paper, the author introduces multiple Cohen strongly \(p\)-summing multilinear operators and polynomials and shows that these ideals are coherent and compatible, see [D. Pellegrino and J. Ribeiro [Monatsh. Math. 173, No. 3, 379–415 (2014; Zbl 1297.46034), arXiv:1101.1992]. In the last section, the correlation between this ideal of polynomials and holomorphy types defined in [L. Nachbin, Topology on spaces of holomorphic mappings. Berlin-Heidelberg-New York: Springer-Verlag (1969; Zbl 0172.39902)] is demonstrated in line with [G. Botelho et al., Stud. Math. 177, No. 1, 43–65 (2006; Zbl 1112.46038)].
The paper is self-contained and the proofs are represented in detail, therefore easy to read.

47L22 Ideals of polynomials and of multilinear mappings in operator theory
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
Full Text: DOI
[1] DOI: 10.1007/BF01427941 · Zbl 0233.47019 · doi:10.1007/BF01427941
[2] Pietsch A, Studia Math 28 pp 333– (1967)
[3] Pietsch A, Operator ideals (1980)
[4] DOI: 10.1002/mana.200310087 · Zbl 1042.47041 · doi:10.1002/mana.200310087
[5] DOI: 10.2989/16073606.2011.640459 · Zbl 1274.47001 · doi:10.2989/16073606.2011.640459
[6] DOI: 10.4064/sm177-1-4 · Zbl 1112.46038 · doi:10.4064/sm177-1-4
[7] DOI: 10.1002/mana.200610791 · Zbl 1181.47076 · doi:10.1002/mana.200610791
[8] DOI: 10.1007/s13348-010-0025-5 · Zbl 1267.47091 · doi:10.1007/s13348-010-0025-5
[9] DOI: 10.1007/s00605-010-0255-3 · Zbl 1236.47062 · doi:10.1007/s00605-010-0255-3
[10] DOI: 10.1016/j.jmaa.2006.04.065 · Zbl 1121.47013 · doi:10.1016/j.jmaa.2006.04.065
[11] Mezrag L, Bull. Belgian Math. Soc. Simon Stevin 16 pp 1– (2009)
[12] DOI: 10.1016/j.jmaa.2012.12.005 · Zbl 1275.47117 · doi:10.1016/j.jmaa.2012.12.005
[13] DOI: 10.1007/978-3-642-88511-2 · doi:10.1007/978-3-642-88511-2
[14] DOI: 10.1016/j.jmaa.2009.10.025 · Zbl 1193.46026 · doi:10.1016/j.jmaa.2009.10.025
[15] DOI: 10.1016/j.jmaa.2010.08.019 · Zbl 1210.47047 · doi:10.1016/j.jmaa.2010.08.019
[16] DOI: 10.1016/j.aim.2011.09.014 · Zbl 1248.47024 · doi:10.1016/j.aim.2011.09.014
[17] Matos MC, Collect. Math 54 pp 111– (2003)
[18] DOI: 10.1016/S0022-247X(03)00352-4 · Zbl 1044.46037 · doi:10.1016/S0022-247X(03)00352-4
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.