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The Index and Other Properties of Elliptic Operators

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

Abstract

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.

Keywords

Vector Bundle Dirac Operator Zero Mode Elliptic Operator Gauge Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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