The Index and Other Properties of Elliptic Operators

  • Albert S. Schwarz
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 307)


We now enumerate, mostly without proof, the most important properties of elliptic operators. Let A be an elliptic operator on a compact manifold M, that is, an elliptic operator on the space of sections Γ = Γ(ξ) of a vector bundle ξ = (E, M, R n , p) over M. The kernel of A, denoted Ker A, is the set of solutions of the equation Af = 0; the dimension l(a) of Ker A is also called the number of zero modes of A. It can be proved that all eigenvalues of A have finite multiplicity, and in particular that l(A) is finite. It can also be shown that the image of A, that is, the set of gΓ such that g = Af for some fG, is a subspace of finite codimension r(A) in Γ; in other words, that set is the space of solutions of a finite set of linear equations.


Vector Bundle Dirac Operator Zero Mode Elliptic Operator Gauge Field 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Albert S. Schwarz
    • 1
  1. 1.Department of Mathematics, 565 Kerr HallUniversity of CaliforniaDavisUSA

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