Lip-Linear operators and their connection to Lipschitz tensor products

TL;DR

This paper introduces Lip-Linear operators acting on Lipschitz tensor products, establishing their properties, connections to bilinear operators, and extending summability concepts like integral and dominated operators.

Contribution

It defines Lip-Linear operators, links them to classical operator spaces, and extends summability theories within this new framework.

Findings

Identification of Lip-Linear operators with classical operator spaces

Extension of summability concepts to Lip-Linear operators

Connections established between Lipschitz tensor products and bilinear operators

Abstract

The linear operators defined on the Lipschitz projective tensor product of X and E motivate the study of a distinct class of operators acting on the cartesian produc X E. This class, denoted by LipL(X E;F), combines Lipschitz and linear properties, forming an intermediate framework between bilinear operators and two-Lipschitz operators. We establish an identification between this space and L(X E;F), which also links it to the space of bilinear operators B(AE(X) E;F). Furthermore, we extend summability concepts within this category, with a particular focus on integral and dominated (p;q)-summing operators