On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators

TL;DR

This paper investigates the approximation properties of unconditionally p-compact and Sinha-Karn p-compact operators in Banach spaces, revealing new relationships and negative answers to existing open problems for 1 ≤ p < 2.

Contribution

It establishes new results on the $ ext{I}$-approximation property for unconditionally p-compact operators and provides counterexamples that answer a problem posed by Kim (2017).

Findings

The $ ext{K}_{u1}$-approximation property does not imply the $ ext{SK}_1$-approximation property.

The $ ext{SK}_1$-approximation property does not imply the $ ext{K}_{u1}$-approximation property.

Existence of subspaces in $ ext{l}^q$ failing all $ ext{SK}_r$-approximation properties for r ≥ p.

Abstract

We establish new results on the $\mathcal I$-approximation property for the Banach operator ideal $\mathcal I=\mathcal{K}_{up}$ of the unconditionally $p$-compact operators in the case of $1\le p<2$. As a consequence of our results, we provide a negative answer for the case $p=1$ of a problem posed by J.M. Kim (2017). Namely, the $\mathcal K_{u1}$-approximation property implies neither the $\mathcal{SK}_1$-approximation property nor the (classical) approximation property; and the $\mathcal{SK}_1$-approximation property implies neither the $\mathcal{K}_{u1}$-approximation property nor the approximation property. Here $\mathcal{SK}_p$ denotes the $p$-compact operators of Sinha and Karn for $p\ge 1$. We also show for all $2<p,q<\infty$ that there is a closed subspace $X\subset\ell^q$ that fails the $\mathcal{SK}_r$-approximation property for all $r\ge p$.