  Suppose $T$ and $S$ are bounded adjointable operators with close range
between Hilbert C*-modules, then $TS$ has closed range if and only if
$Ker(T)+Ran(S)$ is an orthogonal summand, if and only if $Ker(S^*)+Ran(T^*)$ is
an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between
$Ran(S)$ and $Ker(T) \cap [Ker(T) \cap Ran(S)]^{\perp}$ is positive and $
\bar{Ker(S^*)+Ran(T^*)} $ is an orthogonal summand then $TS$ has closed range.
