Fredholm index - Motivation behind it. I have a question concerning the motivation behind the Fredholm index: What is it good for? 
I know that there are many theorems dealing with it, for example that it is continuous, invariant under compact perturbations and so on.
But why are these properties interesting? What does it tell me about the solvability of equations, for example?
Kind regards.
 A: Fredholm was looking at integral equations coming from differential operators. One consequence is that for operators like $\Delta$ on finite domains, you end up with finite-dimensional eigenspaces, and finite deficiencies. And adding relatively compact perturbation in the form a bounded potential $-\Delta +V$, this property is maintained. That already is saying a lot in terms of sovlability. These things are useful in Quantum Mechanics in order to deal with bound states and stability of continuous spectrum as well. The Fredholm index is of theoretical importance for operator algebras. The Fredholm index through pseudo differential operator algebras gives topological information about the manifold itself, which relates solvability to topology.
Here's an interesting way to better understand what Fredholm was up to as well:
https://www.encyclopediaofmath.org/index.php/Fredholm_theorems
The solvability of an integral equation is related to the solvability of the adjoint integral equation, and that's a direct application of the index in its oldest context.
