The difference between the dimensions of the kernel and the cokernel of a Fredholm operator is called its index
A standard equivalent characterization of Fredholm operators is the following:
The image (range) of a Fredholm operator is closed.
The subspace of Fredholm operators in the space of bounded linear operators with the norm topology is open.
In other words, given a Fredholm operator , there exists such that every bounded linear operator satisfying is Fredholm. Fredholm operators on a fixed separable Hilbert space form a semigroup with respect to the composition and the index is a morphism of semigroups: .
Fredholm operators generalize to Fredholm complexes. A finite chain complex
of Banach spaces and bounded operators is said to be a Fredholm complex if the images are closed and the chain homology of the complex is finite dimensional. The Euler characteristic (the alternative sum of the dimensions of the homology groups) is then called the index of the Fredholm complex. Each Fredholm operator can be considered as a Fredholm complex concentrated at zero. Each Fredholm complex produces a Fredholm operator from the sum of the even- to the sum of the odd-numbered spaces in the complex.
One can consider Fredholm almost complexes, where is not zero but the image of that operator is compact. Out of every Fredholm almost complex one can make a complex by correcting the differentials by compact perturbation terms. Elliptic complexes give examples of Fredholm complexes.
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