Strictly singular operators in asymptotic $\ell_{p}$ Banach spaces

Abstract

We present a condition on higher order asymptotic behavior of basic sequences in a Banach space ensuring the existence of bounded noncompact strictly singular operators on a subspace. Applications concern asymptotic $\ell_{p}$ spaces, $1\leq p<\infty$, in particular convexified mixed Tsirelson spaces and related asymptotic $\ell_{p}$ HI spaces.