K-homology and Fredholm operators I: Dirac operators

Relative K-homology and Normal Operators

Let X be a compact metric space. By results of Brown, Douglas and Fillmore, [BDF2], the K-homology of X is realized by Ext(X), the equivalence classes of unital and essential extensions of C(X) by the compact operators K on a separable infinite dimensional Hilbert space H , or equivalently, the equivalence classes of unital and injective ∗-homomorphisms C(X) → Q, where Q = L(H)/K is the Calkin ...

Notes on Fredholm operators

(2) If K ∈ B(X) is compact, then for all λ ∈ C \ {0}, K − λ1 is Fredholm with index zero. (3) The shift operator S± ∈ B(`p) for 1 ≤ p ≤ ∞ defined by (S±x)n = xn±1 is Fredholm with index ±1. (4) If X,Y are finite dimensional and T ∈ B(X,Y ), then by the Rank-Nullity Theorem, ind(T ) = dim(X)− dim(Y ). Lemma 3. Suppose E,F ⊆ X are closed subspaces with F finite dimensional. (1) The subspace E + F...

Subelliptic Spin C Dirac operators, I

Let X be a compact Kähler manifold with strictly pseudoconvex boundary, Y. In this setting, the SpinC Dirac operator is canonically identified with ∂̄ + ∂̄∗ : C∞(X ; Λ) → C∞(X ; Λ). We consider modifications of the classical ∂̄-Neumann conditions that define Fredholm problems for the SpinC Dirac operator. In part 2, [7], we use boundary layer methods to obtain subelliptic estimates for these bound...

Dirac Operators

Dirac Operators and Boundaries

Notes from my lectures in Fall 2003 in MIT course 18.158.