A note on universal operators between separable Banach spaces

TL;DR

This paper compares two recent constructions of universal operators on separable Banach spaces, establishing their isometric equivalence under certain conditions and analyzing their uniqueness and genericity.

Contribution

It demonstrates the isometric equivalence of two universal operators when the target space is the Gurarii space and proves the isometric uniqueness of the operator $P_S$ for fixed $S$.

Findings

Both operators are isometric when $S$ is the Gurarii space.

The operator $P_S$ is uniquely determined up to isometry for fixed $S$.

The operator $oldsymbol{ ext{ extOmega}}$ is generic in a natural infinite game sense.

Abstract

We compare two types of universal operators constructed relatively recently by Cabello S\'anchez, and the authors. The first operator $\Omega$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $\Omega$ is generic in the sense of a natural infinite game.