Lipschitz 𝑝-summing multilinear operators correspond to Lipschitz 𝑝-summing operators

Abstract

We give conditions that ensure that an operator satisfying a Piestch domination in a given setting also satisfies a Piestch domination in a different setting. From this we derive that a bounded multilinear operator T T is Lipschitz p p -summing if and only if the mapping f T ( x 1 ⊗ ⋯ ⊗ x n ) ≔ T ( x 1 , … , x n ) f_T(x_1\otimes \cdots \otimes x_n)≔T(x_1,\ldots , x_n) is Lipschitz p p -summing. The results are based on the projective tensor norm. An example with the Hilbert tensor norm is provided to show that the statement may not hold when a reasonable cross-norm other than the projective tensor norm is considered.