The Giesy--James theorem for general index $p$, with an application to operator ideals on the $p$th James space

TL;DR

This paper extends the Giesy--James theorem to the $p$th James space for all $p eq 1$, and applies it to identify a new closed ideal of operators on these spaces, enriching the understanding of their operator structure.

Contribution

It generalizes the Giesy--James theorem to all $p$ in (1,∞) and introduces a new closed ideal of operators on $J_p$, expanding the theory of operator ideals.

Findings

$c_0$ is finitely representable in $J_p$ for all $p eq 1$

Identifies a new closed ideal of operators on $J_p$

Provides insights into the structure of operator ideals on James spaces

Abstract

A theorem of Giesy and James states that $c_0$ is finitely representable in James' quasi-reflexive Banach space $J_2$. We extend this theorem to the $p$th quasi-reflexive James space $J_p$ for each $p \in (1,\infty)$. As an application, we obtain a new closed ideal of operators on $J_p$, namely the closure of the set of operators that factor through the complemented subspace $(\ell_\infty^1 \oplus \ell_\infty^2 \oplus...\oplus \ell_\infty^n \oplus...)_{\ell_p}$ of $J_p$.