We give sufficient conditions on an asymptotic $\ell_p$ (for $1 < p < \infty$) Banach space to 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 noncompact operators on the HI spaces constructed by G. Androulakis and the author and by Deliyanni and 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 $X$.
"Operators on asymptotic $\ell_p$ spaces which are not compact perturbations of a multiple of the identity." Illinois J. Math. 52 (2) 515 - 532, Summer 2008. https://doi.org/10.1215/ijm/1248355347