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

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 ${\mathcal{S}\mathcal{K}}_1$-approximation property nor the (classical) approximation property; and the ${\mathcal{S}\mathcal{K}}_1$-approximation property implies neither the ${\mathcal{K}}_{u1}$-approximation property nor the approximation property. Here ${\mathcal{S}\mathcal{K}}_p$ 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 ${\mathcal{S}\mathcal{K}}_r$-approximation property for all $r\ge p$.

23 pages

References in corpus (2)

• Closed ideals in the algebra of compact-by-approximable operators
• Quotient algebras of Banach operator ideals related to non-classical approximation properties