The ideal of p-compact operators: a tensor product approach

TL;DR

This paper explores the structure of p-compact operators using tensor norms, establishing their relation to p-summing operators and revealing key properties and dualities within operator ideals.

Contribution

It introduces a tensor product approach to p-compact operators, linking them to Chevet-Saphar tensor norms and characterizing their structural properties and dual relationships.

Findings

$ ext{K}_p$ is associated to $/d_p$, the left injective associate of $d_p$

$ ext{K}_p(E;F)$ equals $ ext{K}_q(E;F)$ for certain p, q ranges

$ ext{K}_p$ is regular, surjective, totally accessible, and dual to $ ext{Pi}_p$

Abstract

We study the space of $p$-compact operators $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet-Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators with that of $p$-compact operators. With the results known for the former class and appropriate hypothesis on $E$ and $F$ we prove that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We also exhibit several structural properties of $\mathcal K_p$. For instance, we obtain that $\mathcal K_p$ is regular, surjective, totally accessible and characterize its maximal hull $\mathcal K_p^{max}$ as the dual ideal of the $p$-summing operators, $\Pi_p^{dual}$. Furthermore, we prove that $\mathcal K_p$ coincides isometrically with $\mathcal {QN}_p^{dual}$, the dual ideal of the quasi $p$-nuclear operators.