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Grothendieck's theorem for absolutely summing multilinear operators is optimal

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Pellegrino, Daniel and Seoane-Sepúlveda, Juan B. (2015) Grothendieck's theorem for absolutely summing multilinear operators is optimal. Linear & Multilinear Algebra, 63 (3). pp. 554-558. ISSN 0308-1087

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Official URL: http://www.tandfonline.com/doi/abs/10.1080/03081087.2013.877013#.VTDkqvmsWCl




Abstract

Grothendieck's theorem asserts that every continuous linear operator from ℓ1 to ℓ2 is absolutely (1;1)-summing. In this note we prove that the optimal constant gm so that every continuous m-linear operator from ℓ1×⋯×ℓ1 to ℓ2 is absolutely (gm;1)-summing is 2m+1. We also show that if gm<2m+1 there is c dimensional linear space composed by continuous non absolutely (gm;1)-summing m-linear operators from ℓ1×⋯×ℓ1 to ℓ2. In particular, our result solves (in the positive) a conjecture posed by A.T. Bernardino in 2011.


Item Type:Article
Uncontrolled Keywords:lineability, spaceability, absolutely summing operators
Subjects:Sciences > Mathematics
ID Code:29656
Deposited On:17 Apr 2015 11:23
Last Modified:25 Nov 2016 12:38

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