Abstract. Given Hilbert space operators $A$ and $B$, we characterize solvability of the operator equation $A = BJ + G$, where $J$ is a Fredholm operator of prescribed index and $G$ is an operator of prescribed finite rank. In the case when $A$ or $B$ has closed range, we characterize solvability of $A = BJ + K$, with $J$ Fredholm and $K$ compact. More generally, for elements $a$, $b$ of a $C^*$-algebra ${\cal A}$, we study solvability of the equation $a = bj$, $j$ invertible. There are natural majorization and annihilator conditions necessary for solvability, and we show that if ${\cal A}$ is Rickart, then these conditions are also sufficient.

AMS Subject Classification (1991): 47A68, 47A62

Received February 10, 1993. (Registered under 5575/2009.)