On strictly singular operators between separable Banach spaces

TL;DR

This paper introduces a new ordinal rank for strictly singular operators between separable Banach spaces, compares it with existing ranks, and constructs specific Banach spaces to analyze the complexity of these operators.

Contribution

It defines a new co-analytic rank on strictly singular operators, shows it dominates previous ranks, and constructs Banach spaces with complex operator sets where strict singularity is bounded.

Findings

The new rank $ s$ is co-analytic and dominates $ ho$.

Constructed spaces $Y_p$ with complex sets of strictly singular operators.

Every strictly singular operator from $oldsymbol{ ext{ell}}_p$ to $Y_p$ has $ ho$-rank at most 2.

Abstract

Let $X$ and $Y$ be separable Banach spaces and denote by $\sss\sss(X,Y)$ the subset of $\llll(X,Y)$ consisting of all strictly singular operators. We study various ordinal ranks on the set $\sss\sss(X,Y)$. Our main results are summarized as follows. Firstly, we define a new rank $\rs$ on $\sss\sss(X,Y)$. We show that $\rs$ is a co-analytic rank and that dominates the rank $\varrho$ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math., 169 (2009), 221-250]. Secondly, for every $1\leq p<+\infty$ we construct a Banach space $Y_p$ with an unconditional basis such that $\sss\sss(\ell_p, Y_p)$ is a co-analytic non-Borel subset of $\llll(\ell_p,Y_p)$ yet every strictly singular operator $T:\ell_p\to Y_p$ satisfies $\varrho(T)\leq 2$. This answers a question of Argyros.