  Using the notion of $S_\xi$-strictly singular operator introduced by
Androulakis, Dodos, Sirotkin and Troitsky, we define an ordinal index on the
subspace of strictly singular operators between two separable Banach spaces. In
our main result, we provide a sufficient condition implying that this index is
bounded by $\omega_1$. In particular, we apply this result to study operators
on totally incomparable spaces, hereditarily indecomposable spaces and spaces
with few operators.
