On the closedness of the sum of ranges of operators with almost compact products

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