If a differential operator on a smooth Hermitian vector bundle over a compact manifold is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If is also elliptic, then the Hilbert space of square integrable sections of with the canonical left -action and the operator for a normalizing function is a Fredholm module, and its -homology class is independent of . 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.
"Elliptic operators and K-homology." Rocky Mountain J. Math. 50 (1) 91 - 124, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.91