Operators on Asymptotic $\ell_p$ Spaces which are not Compact Perturbations of a Multiple of the Identity

TL;DR

This paper establishes conditions under which asymptotic $ ext{ell}_p$ spaces admit non-compact perturbations of scalar multiples of the identity, revealing new operator structures and embedding properties.

Contribution

It provides sufficient conditions for the existence of non-compact, non-compact perturbation operators on certain asymptotic $ ext{ell}_p$ spaces and shows $ ext{ell}_$ embeds into their operator spaces.

Findings

Existence of non-compact operators under specified conditions.

Embedding of $ ext{ell}_$ into the operator space of $X$.

Application to HI spaces constructed by previous authors.

Abstract

We give sufficient conditions on an asymptotic $\ell_p$ (for $1 < p < \infty$) Banach space which ensure the space admits an operator which is not a compact perturbation of a multiple of the identity. These conditions imply the existence of strictly singular non-compact operators on the HI spaces constructed by G. Androulakis and the author and by I. Deliyanni and A. Manoussakis. Additionally we show that under these same conditions on the space $X$, $\ell_\infty$ embeds isomorphically into the space of bounded linear operators on $X$.