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∗)+Ran(T∗) is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between Ran(S) and Ker(T)∩[Ker(T)∩Ran(S)]⊥ is positive and is an orthogonal summand then TS has closed range.