Strictly singular non-compact operators on a class of HI spaces
  We present a method for constructing bounded strictly singular non-compact operators on mixed Tsirelson spaces defined either by the families (A_n) or (S_n) of a certain class, as well as on spaces built on them, including hereditarily indecomposable spaces. 
Antonis Manoussakis, Anna Pelczar-Barwacz
math.FA