Projection from space of \emph{two-Lipschitz} operators onto the space of Bilinear maps

TL;DR

This paper investigates the projection of two-Lipschitz operators onto bilinear maps, establishing existence, isometric isomorphisms, and conditions for bilinearity, with applications to dual space characterizations.

Contribution

It introduces a norm-one projection from two-Lipschitz operators to bilinear maps under certain conditions and characterizes when two-Lipschitz operators are bilinear.

Findings

Existence of a norm-one projection using invariant means.

Isometric isomorphism of a quotient space with a specific operator space.

Necessary and sufficient conditions for two-Lipschitz operators to be bilinear.

Abstract

In this article, we establish the existence of a norm-one projection from the space of all \emph{two-Lipschitz} operators onto the space of all bounded bilinear operators under certain conditions on the corresponding codomain spaces, using the method of invariant means. We also show that, when the codomain is an injective Banach space, the quotient of the \emph{two-Lipschitz} operator space by the bounded bilinear space is isometrically isomorphic to a specific operator space, via vector-valued duality. We conclude by proving a necessary and sufficient condition for a \emph{two-Lipschitz} operator to be a bilinear map. As an application of the theory developed here, we present an alternative proof that $\bigslant{L^{\infty}(\mathbb{R}\times \mathbb{R})}{span\{\textbf{1}\}}$ is a dual space.