The product of operators with closed range in Hilbert C*-modules

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 $Ker(S^{\ast })+Ran(T^{\ast })$ 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 $\overline{Ker(S^{\ast })+Ran(T^{\ast })}$ is an orthogonal summand then $TS$ has closed range.