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Weakly compact and absolutely summing polynomials. (English) Zbl 1036.46034
Let $$E$$ be an infinite dimensional Banach space. It is well known that every absolutely summing linear operator between Banach spaces is weakly compact. In this paper, in contrast to the linear case, the author shows that for each $$n$$, $$n \geq 2,$$ there is a continuous $$n$$-homogeneous polynomial between Banach spaces which is absolutely $$n$$-summming but is not weakly compact. The author also shows that there is a continuous $$n$$-homogeneous polynomial $$P: E \to E$$ which plays the role of the identity operator, that means, $$P$$ is neither $$r$$-summing for any $$r$$ nor compact and $$P$$ is weakly compact, if and only if $$E$$ is reflexive.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47H60 Multilinear and polynomial operators
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##### References:
 [1] Alencar, R.; Matos, M., Some classes of multilinear mappings between Banach spaces, Pub. dep. anal. mat. univ. complut. Madrid, 12, (1989) [2] Botelho, G., Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. roy. irish acad. sect. A, 97, 145-153, (1997) · Zbl 0903.46018 [3] Botelho, G., Almost summing polynomials, Math. nachr., 211, 25-36, (2000) · Zbl 0960.46029 [4] Defant, A.; Floret, K., Tensor norms and operator ideals, (1993), North-Holland Amsterdam · Zbl 0774.46018 [5] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators, (1995), Cambridge Univ. Press Cambridge · Zbl 0855.47016 [6] Dineen, S., Complex analysis on infinite dimensional spaces, (1999), Springer-Verlag London · Zbl 1034.46504 [7] Floret, K.; Matos, M., Application of a Khintchine inequality to holomorphic mappings, Math. nachr., 176, 65-72, (1995) · Zbl 0839.46040 [8] Matos, M., Absolutely summing holomorphic mappings, An. acad. brasil ciênc., 68, 1-13, (1996) · Zbl 0854.46042 [9] M. Matos, The Dvoretzky-Rogers theorem for polynomial and multilinear mappings, in45o. Seminário Brasileiro de Análise, Florianópolis, 1997, pp. 849-856. [10] Y. Meléndez, and, A. Tonge, Absolutely summing polynomials, preprint. · Zbl 0338.46040 [11] Pietsch, A., Ideals of multilinear functionals, Proceedings of the second international conference on operator algebras, ideals and their applications in theoretical physics, (1983), Teubner Leipzig, p. 185-199 [12] Schneider, B., On absolutely p-summing and related multilinear mappings, Wiss. Z. brandenburg. landeshochsch., 35, 105-117, (1991) · Zbl 0777.47016
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