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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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