Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

We show the existence of Banach spaces X, Y such that the set of strictly singular operators (X,Y) (resp., the set of strictly cosingular operators (X,Y)) would be strictly included in ℱ+(X,Y) (resp., ℱ−(X,Y)) for the nonempty class of closed densely defined upper semi-Fredholm operators Φ+(X,Y) (resp., for the nonempty class of closed densely defined lower semi-Fredholm operators Φ−(X,Y))