  For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we
prove that the Hodge theory holds for $A$-elliptic complexes of
pseudodifferential operators acting on smooth sections of finitely generated
projective $A$-Hilbert bundles over $M.$ For these $C^*$-algebras, we get also
a topological isomorphism between the cohomology groups of an $A$-elliptic
complex and the space of harmonic elements. Consequently, the cohomology groups
appear to be Banach spaces. We prove as well, that if the Hodge theory holds
for a complex in the category of Hilbert $A$-modules and continuous adjointable
Hilbert $A$-module homomorphisms, the complex is self-adjoint parametrix
possessing.
