Strictly singular operators in asymptotic ${\ell }_p$ Banach spaces

Abstract

We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic ${\ell }_p$ spaces, $1\le p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic ${\ell }_p$ HI spaces.

arXiv comment19 pages