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

TL;DR

This paper establishes an equivalence between Lipschitz p-summing multilinear operators and Lipschitz p-summing operators via the projective tensor norm, highlighting the importance of the tensor norm choice.

Contribution

It proves that Lipschitz p-summing multilinear operators correspond exactly to Lipschitz p-summing operators under the projective tensor norm, with counterexamples for other norms.

Findings

Equivalence holds for the projective tensor norm

Counterexample with Hilbert tensor norm shows the equivalence may fail

Provides conditions for Piestch domination transfer

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 mutlilinear operator $T$ is Lipschitz $p$-summing if and only if the mapping $ f_T(x_1\otimes\cdots \otimes x_n):=T(x_1,\ldots, x_n)$ is Lipschitz $p$-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.