On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators

Abstract

Abstract. A famous result due to Grothendieck asserts that every continuous linear operator from ℓ 1 to ℓ 2 is absolutely (1, 1)-summing. If n ≥ 2, however, it is very simple to prove that every continuous n-linear operator from ℓ 1 × · · · × ℓ 1 to ℓ 2 is absolutely (1; 1, ..., 1)-summing, and even absolutely 2 n ; 1, ..., 1 -summing. In this note we deal with the following problem:Given a positive integer n ≥ 2, what is the best constant g n > 0 so that every n-linear operator from ℓ 1 × · · · × ℓ 1 to ℓ 2 is absolutely (g n ; 1, ..., 1)-summing?We prove that g n ≤ 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.

Sharp coincidences for absolutely summing multilinear operators

“…Very recently the following result was essentially proved independently by different authors (see [6,Thm 1.4] and also [8,49]):…”

“…A result due to G. Botelho [10] asserts that, under certain cotype assumptions, the Defant-Voigt theorem (1.7) can be improved even with arbitrary Banach spaces F in the place of the scalar field K: if n ≥ 2 is a positive integer, s ∈ [2, ∞) and F = {0} is any Banach space, then (1.8) inf r : L(E 1 , ..., E n ; F ) = as(r;1,...,1) (E 1 , ..., E n ; F ) for all E j in C (s) ≤ s n and the value r = s n is attained. Using recent results (see [6,8,49]) it is also simple to conclude that sup r : L(E 1 , ..., E n ; F ) = as(1;r,...,r) (E 1 , ..., E n ; F ) for all E j in C (s) ≥ sn sn + s − n and sup r : L(E 1 , ..., E n ; F ) = as(2;r,...,r) (E 1 , ..., E n ; F ) for all E j in C (s) ≥ 2sn 2sn + s − 2n , with both r = sn sn+s−n and r = 2sn 2sn+s−2n attained. Several recent papers have treated similar problems involving inclusion, coincidence results and the geometry of the Banach spaces involved (see [12,13,29,38,49]).…”

“…In the nonlinear setting ℓ 1 also plays a special role. For example, the following multilinear extension of Grothendieck's Theorem was recently proved in [6]: L( n ℓ 1 ; ℓ 2 ) = as( 2 n+1 ;1,...,1) ( n ℓ 1 ; ℓ 2 ). Note that the case n = 1 is precisely Grothendieck's Theorem.…”

On the transformation of vector-valued sequences by linear and multilinear operators

“…It is easy to check that the sequence classes ℓ ∞ (·), c 0 (·) and ℓ p (·), 1 ≤ p ≤ +∞, are multilinearly stable (see (5)).…”

Composition results for strongly summing and dominated multilinear operators

“…Without any claim of completeness we mention: absolutely summing multilinear operators, see [1,2,9,24,37]; multiple summing multilinear operators, see [3,8,28,32]; dominated multilinear operators, see [21,27,30,31,33]. For the theory of polynomials in Banach spaces and their applications the interested reader can consult [17,18,29] and for the connection between holomorphy types and ideals of multilinear mappings we recommend the reader [7].…”