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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.

46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
Full Text: DOI
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