Abstract. In this paper it is shown that if $A$ and $B$ are closed range operators in a Hilbert space for which the equation $B=XA$ has at least one solution, then the compactness of $A-B$ is equivalent to the existence of a solution $X$ such that $X-I$ is compact. This result has several consequences on the description of the compact perturbations of particular classes of operators.

AMS Subject Classification (1991): 47A55, 47A05

Received March 23, 2005, and in revised form August 2, 2005. (Registered under 5924/2009.)