Descriptive set theoretic methods applied to strictly singular and   strictly cosingular operators
  The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of $\omega_1$ subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied. 
G. Androulakis, K. Beanland