Fredholm index

Let P be a Fredholm operator. The index of P is defined as

index(P) = dimker(P)-dimcoker(P)
= dimker(P)-dimker(P*).

Note: this is well defined as ker(P) and ker(P*) are finite-dimensional vector spacesMathworldPlanetmath, for P Fredholm.

  • index(P*)=-index(P).

  • index(P+K)=index(P) for any compact operatorMathworldPlanetmath K.

  • If P1:12 and P2:23 are Fredholm operators, then index(P2P1)=index(P1)+index(P2).

  • If tPt, t[0,1] is a norm continuousMathworldPlanetmath path of Fredholm operators, then index(Pt)=index(P0).

Fredholm operators of the form 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒+𝑐𝑜𝑚𝑝𝑎𝑐𝑡 have index zero.

Title Fredholm index
Canonical name FredholmIndex
Date of creation 2013-03-22 13:20:45
Last modified on 2013-03-22 13:20:45
Owner mhale (572)
Last modified by mhale (572)
Numerical id 9
Author mhale (572)
Entry type Definition
Classification msc 47A53
Synonym index
Related topic FredholmOperator