2011/02/22 by A. Thiago Lopes Bernardino, Bernardino, A. Thiago Lopes
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1102.4542
6 pages
arxiv created 2011/03/18 · arxiv updated 2011/03/21
A famous result due to Grothendieck asserts that every continuous linear operator from ℓ1 to ℓ2 is absolutely (1,1)-summing. If n≥2, however, it is very simple to prove that every continuous n-linear operator from ℓ1×...×ℓ1 to ℓ2 is absolutely (1;1,...,1) -summing, and even absolutely (\frac2% n;1,...,1) -summing. In this note we deal with the following problem: Given a positive integer n≥2, what is the best constant gn>0 so that every n-linear operator from ℓ1×...×ℓ1 to ℓ2 is absolutely (gn;1,...,1) -summing? We prove that gn≤(2)/(n+1) and also obtain an optimal improvement of previous recent results (due to Heinz Juenk et al, Geraldo Botelho et al and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.