Here we give a survey of the properties of the map R: if it has dense range or closed range, if it is surjective, etc., and describe some applications. We present examples showing that the properties of R can be very different on different spaces E. In some cases the only compact operator in the image of R is the null operator, in the other cases R is surjective, and in the case of , where J is James' space, we have that and the image of R is the class of lattice regular operators on . Among the applications, we show how to obtain examples of tauberian operators T so that is not tauberian, and operators such that R(T) is invertible in but T fails to be invertible modulo W(E).