Accessible operators on ultraproducts of Banach spaces

TL;DR

This paper explores the nature of linear operators on ultraproducts of Banach spaces, linking them to twisted sums and quasilinear maps, and provides concrete examples for sequence spaces.

Contribution

It establishes a connection between accessible operators on ultraproducts and twisted sums, offering new insights and examples in Banach space theory.

Findings

Accessible operators can exist without being ultraproducts of operators.

A correspondence between accessible operators and non-splitting short exact sequences.

Concrete examples of accessible functionals for sequence spaces $ ext{ell}_p$.

Abstract

We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps. We provide a bridge between these "accessible" operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces $X$ and $Y$, there is an "accessible" operator $X_U\to Y_U$ that is not the ultraproduct of a family of operators $X\to Y$ if and only if there is a short exact sequence of quasi-Banach spaces and operators $0\to Y\to Z\to X\to 0$ that does not split. We then adapt classical work by Ribe and Kalton--Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces $\ell_p$. The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory.