A Hilbert bundle description of differential K-theory

We give an infinite dimensional description of the differential K-theory of a manifold $M$. The generators are triples $[H,A,\omega ]$ where $H$ is a ${\bf Z}_2$-gradedHilbertbundleon$M$,$A$isasuperconnectionon$H$and$\omega$isadifferentialformon$M$. The relations involve eta forms. We show that the ensuing group is the differential K-group ${\check{K}}^0(M)$. In addition, we construct the pushforward of a finite dimensional cocycle under a proper submersion with a Riemannian structure. We give the analogous description of the odd differential K-group ${\check{K}}^1(M)$. Finally, we give a model for twisted differential K-theory.