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

TL;DR

This paper extends classical operator ideals to Lip-linear operators and provides a characterization of Hilbert spaces based on these operators, generalizing Kwapień's theorem.

Contribution

It introduces Lip-linear operator classes derived from $p$-summing ideals and characterizes Hilbert spaces via these classes.

Findings

Characterization of Hilbert spaces using Lip-linear operators

Extension of $p$-summing and strongly $p$-summing operators to Lip-linear context

Equivalence of certain Lip-linear operator properties with Hilbert space structure

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.