The linear operators defined on the Lipschitz projective tensor product \(X {\widehat{\boxtimes }}_{\pi }E\) motivate the study of a distinct class of operators acting on the cartesian product \(X\times E\) . These operators, called Lip-linear operators, form a Banach space denoted by \(LipL_{0}\left( X\times E;F\right) .\) This space provides an intermediate setting between bilinear operators and two-Lipschitz operators. We establish a natural identification between \(LipL_{0}\left( X\times E;F\right) \) and \({\mathcal {L}} (X{\widehat{\boxtimes }}_{\pi }E;F) ,\) which also relates it to the space of bilinear operators \({\mathcal {B}}\left( {\mathcal {F}}(X)\times E;F\right) \) . Furthermore, we extend summability concepts within this category, with a particular focus on integral and dominated (p; q)-summing operators.
