Sharp coincidences for absolutely summing multilinear operators

Abstract

Abstract. In this note we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators. We connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar results obtained via the complex interpolation method.

Optimal estimates for summing multilinear operators

“…The fact that 2m 2m−1 can replace km km−1 in all cases ensures that s = km km−1 is not attained and thus improves the estimate of [22,Corollary 3.1], which can be improved to sup{s :…”

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

“…The consideration of these and other classes of vector-valued sequences originated several well studied ideals of linear and multilinear operators -Banach operator ideals and Banach multi-ideals; which, up to now, have been investigated individually in the literature. The following examples show just how broad in scope this approach is: (i) p-dominated n-linear operators send weakly p-summable sequences to absolutely p nsummable sequences [10,23,32] (the case n = 1 recovers the p-summing linear operators), (ii) absolutely (s; r)-summing linear and multilinear operators send weakly r-summable sequences to absolutely s-summable sequences [5,6,26], (iii) unconditionally p-summing linear and multilinear operators send weakly p-summable sequences to unconditionally p-summable sequences [24], (iv) almost summing linear and multilinear operators send unconditionally summable sequences to almost unconditionally summable sequences [9,27,29], (v) a multilinear operator is weakly sequentially continuous at the origin if it sends weakly null sequences to norm null sequences [3,20,38] (the case n = 1 gives the ideal of completely continuous linear operators).…”