Lipschitz extensions of linear operators

TL;DR

This paper explores conditions under which linear operators between Banach spaces can be extended as Lipschitz maps within certain operator ideals, revealing a deep connection between linear and Lipschitz extensions.

Contribution

It establishes a characterization of linear operator extensions as Lipschitz maps within Banach operator ideals, linking linear and Lipschitz extension theories.

Findings

Linear operators extendable to Lipschitz maps belong to related Lipschitz operator ideals.

Extension criteria are characterized by the operator ideal membership.

Linear factorization through ℓ∞(Γ) is equivalent to Lipschitz factorization.

Abstract

Let $E, F, E_0$ be Banach spaces, with $E_0$ a subspace of $E$. For a maximal Banach operator ideal $\mathcal{A}$, we show that a linear operator from $E_0$ to $F$ can be extended to a linear operator from $E$ to $F$ that belongs to $\mathcal{A}$ if and only if it can be extended to a Lipschitz map from $E$ to $F$ belonging to a wide class of Lipschitz Banach operator ideals related with $\mathcal{A}$. As a consequence, we show that linear operators with special Lipschitz factorization through $\ell_{\infty}(\Gamma)$ has analogous linear factorization through $\ell_{\infty}(\Gamma)$.