Extensions of Grothendieck and Bennett–Carl theorems

Grothendieck has proved that the inclusion operator J : l 1 ↪ l 2 is 1-summing. Bennett and Carl proved, independently, that if 1 ≤ p ≤ q ≤ 2 and 1 s = 1 p - 1 q + 1 2 , then the inclusion operator J : l p ↪ l q is ( s , 1) -summing. In this paper we prove the following extension of Grothendieck the...

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Bibliographic Details
Published in:Revista matemática complutense 2016-09, Vol.29 (3), p.677-690
Main Author: Popa, Dumitru
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Banach spaces
Convergence
Error analysis
Geometry
Mathematics
Mathematics and Statistics
Multiplication
Multiplication & division
Theorems
Topology
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Summary:Grothendieck has proved that the inclusion operator J : l 1 ↪ l 2 is 1-summing. Bennett and Carl proved, independently, that if 1 ≤ p ≤ q ≤ 2 and 1 s = 1 p - 1 q + 1 2 , then the inclusion operator J : l p ↪ l q is ( s , 1) -summing. In this paper we prove the following extension of Grothendieck theorem: If X , Y are Banach spaces and V : X → Y is 1-summing then, the multiplication operator M V : l 1 [ X ] → l 2 ( Y ) is 1 -summing. Using this result we prove the following extension of Bennett and Carl theorem: If X and Y are Banach spaces and V : X → Y is 1-summing then, the multiplication operator M V : l p ( X ) → l q ( Y ) is ( s , 1)-summing. l 1 [ X ] , l p ( X ) denotes the Banach spaces of all unconditionally norm convergent series respectively p -absolutely convergent series and M V ( ( x n ) n ∈ N ) : = ( V ( x n ) ) n ∈ N is the multiplication operator.
ISSN:1139-1138
1988-2807
DOI:10.1007/s13163-016-0198-x