  Let $\mathcal{H}_1,\ldots,\mathcal{H}_n,\mathcal{H}$ be complex Hilbert
spaces and $A_k:\mathcal{H}_k\to\mathcal{H}$ be a bounded linear operator with
the closed range $Ran(A_k)$, $k=1,\ldots,n$. It is known that if $A_i^*A_j$ is
compact for any $i\neq j$, then $\sum_{k=1}^n Ran(A_k)$ is closed.
  We show that if all products $A_i^*A_j$, $i\neq j$ are "almost" compact, then
the subspaces $Ran(A_1),\ldots,Ran(A_n)$ are essentially linearly independent
and their sum is closed.
