Strictly singular operators in asymptotic $\ell_p$ Banach spaces

TL;DR

This paper establishes conditions on the asymptotic structure of Banach spaces that guarantee the existence of bounded, non-compact, strictly singular operators, with applications to asymptotic _p spaces and related constructions.

Contribution

It introduces new criteria linking higher order asymptotic behavior to the existence of strictly singular operators in Banach spaces.

Findings

Conditions for strictly singular operators in asymptotic _p spaces

Application to convexified mixed Tsirelson spaces

Identification of operators in asymptotic _p HI 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, $1\leq p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic $\ell_p$ HI spaces.