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 $T$ 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.