Elliptic operators and K-homology

Abstract

If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of square integrable sections of $S$ with the canonical left $C\left(M\right)$-action and the operator $\chi \left(D\right)$ for $\chi$ a normalizing function is a Fredholm module, and its $K$-homology class is independent of $\chi$. In this expository article, we provide a detailed proof of this fact following the outline in the book “Analytic K-homology” by Higson and Roe.