Lipschitz $p$-summing multilinear operators

TL;DR

This paper develops a new theory of Lipschitz $p$-summing multilinear operators using geometric $\Sigma$-operators, establishing key theorems and characterizations that extend classical results in operator theory.

Contribution

It introduces Lipschitz $p$-summing multilinear operators, providing a comprehensive framework with domination, factorization, and tensorial characterizations, extending classical operator theorems.

Findings

Establishes Pietsch-type domination and factorization theorems for the new class.

Characterizes the class via Chevet-Saphar-type tensor norms.

Relates Lipschitz $p$-summing operators to Dunford-Pettis and Hilbert-Schmidt multilinear operators.

Abstract

We apply the geometric approach provided by $\Sigma$-operators to develop a theory of $p$-summability for multilinear operators. In this way, we introduce the notion of Lipschitz $p$-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pelczy\'nsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type tensor norm. Moreover, we introduce the notion of Dunford-Pettis multilinear operators. With them, we characterize when a projective tensor product contains $\ell_1$. Relations between Lipschitz $p$-summing multilinear operators with Dunford-Pettis and Hilbert-Schmidt multilinear operators are given.