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How can one calculate the index of a Fredholm operator numerically ?

In numerically calculations one uses always finte dimensional spaces. But linear operators on finite dimensional spaces have always index zero.

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If you happen to be lucky enough for the operator to be an (ordinary) differential operator, e.g. $Ly = y' - A(x)y$, and you are even more lucky so that the limits $A(\pm \infty)$ exist and have spectrum away from the imaginary axis, then the Fredholm index of L is the difference between the Morse index of A(\infty) and the Morse index of A(-\infty). –  Aaron Hoffman Aug 24 '11 at 20:03
Are you thinking of operators on particular (function) spaces, or a Fredholm operator in full generality? –  Yemon Choi Aug 24 '11 at 20:50
A linear operator between two DISTINCT finite-dimensional vector spaces does not have index zero, and that may help. –  Alain Valette Aug 25 '11 at 7:27
@Aaron: of course, assuming $A(x)$ is a path of operators on a finite dimensional space. –  Pietro Majer Aug 25 '11 at 7:29
Alain ... but it has non-zero index for a TRIVIAL reason! –  Helge Aug 25 '11 at 11:06
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1 Answer

up vote 2 down vote accepted

The two key properties of the Fredholm index are

  • It is a (norm)-continuous function from the bounded linear operators to the integers. In particular, if $A$ is a Fredholm operator, then there exists $\delta > 0$ such that for $\|A - B\| < \delta$, we have $index(A) = index(B)$. This tells you that you can approximate your problem.
  • The Fredholm index doesn't see compact perturbations. So if $A$ is Fredholm and $K$ is compact, then $index(A +K ) = index(A)$. This tells you that you cannot do naive computations like picking some finite orthonormal set $\psi_{j}$ with $j=1,\dots,N$ and hope that the $N \times N$ matrix $$ A_{j,k} = \langle \psi_j, A \psi_k\rangle $$ tells you anything about the Fredholm index of $A$.

So you will now need to do something smarter. This is possible in many particular cases, for example for Toeplitz operators. The first property allows one to reduce the computation of the index to the computation of the winding number of a polynomial. Or the Atiyah--Singer index theorems reduces computing the index to some topological information ...

So to get a more meaningful answer, you will need to be more specific about the problem.

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For more refined invariance theorems one the Fredholm index check Hoermander's The Analysis of Linear Partial Differential Operators. –  Pietro Majer Aug 25 '11 at 7:32
Which volume? There are 4 if I remember correctly... –  Helge Aug 25 '11 at 11:06
It's Vol III, Chapter 19 (Elliptic Operators on a Compact Manifold Without Boundary), Sec.19.1 Abstract Fredholm Theory. –  Pietro Majer Aug 25 '11 at 12:07
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