On the closedness of the sum of ranges of operators AkAk with almost compact products Ai⁎Aj

Let H1,…,Hn,HH1,…,Hn,H be complex Hilbert spaces and Ak:Hk→HAk:Hk→H be a bounded linear operator with closed range Ran(Ak)Ran(Ak), k=1,…,nk=1,…,n. It is known that if Ai⁎Aj is compact for all i≠ji≠j, then ∑k=1nRan(Ak) is closed. We show that if all the products Ai⁎Aj, i≠ji≠j, are “almost” compact (in a certain sense), then the subspaces Ran(A1),…,Ran(An)Ran(A1),…,Ran(An) are essentially linearly independent and their sum is closed.