On strictly singular operators between separable Banach spaces

Let $X$ and $Y$ be separable Banach spaces and denote by $\text{sss}\text{sss}(X,Y)$ the subset of $\text{llll}(X,Y)$ consisting of all strictly singular operators. We study various ordinal ranks on the set $\text{sss}\text{sss}(X,Y)$. Our main results are summarized as follows. Firstly, we define a new rank $\text{rs}$ on $\text{sss}\text{sss}(X,Y)$. We show that $\text{rs}$ 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 $1\le p<+\mathrm{\infty }$ we construct a Banach space $Y_p$ with an unconditional basis such that $\text{sss}\text{sss}({\mathrm{\ell }}_p,Y_p)$ is a co-analytic non-Borel subset of $\text{llll}({\mathrm{\ell }}_p,Y_p)$ yet every strictly singular operator $T:{\mathrm{\ell }}_p\to Y_p$ satisfies $\varrho (T)\le 2$. This answers a question of Argyros.