Images of elliptic operators on Hilbert bundles on compact manifolds are closed

We prove that the image of an elliptic operator on a smooth separable Hilbert fibre bundle on compact manifolds is closed with respect to the natural pre-Hilbert topology. We consider a tensor product of the operator, which is invariant with respect to an action of the C*-algebra of compact operators, and compare the image of this tensor product with the image of the original operator. This establishes a ground for the Hodge theory for these structures.