  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.
