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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.

MSC:
47L22 Ideals of polynomials and of multilinear mappings in operator theory
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
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