Composition ideals of Lip-Linear operators and a Hilbert space characterization

Abstract

In this paper, we investigate classes of Lip-linear operators constructed using the composition ideal method. We focus on two fundamental linear operator ideals, $p$-summing and strongly $p$-summing operators, and extend them to define the corresponding classes of Lip-linear operators. Several key results are established, including a characterization theorem for Hilbert spaces originally due to Kwapie\'{n}. Specifically, we show that a Banach space $F$ is isomorphic to a Hilbert space if and only if every factorable strongly $p$-summing Lip-linear operator with values in $F$ is Cohen strongly $p$-summing.