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Fredholm index (Definition)

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

$\displaystyle \mathop{\mathrm{index}}\nolimits (P)$ $\displaystyle =$ $\displaystyle \dim\ker(P) - \dim\mathop{\mathrm{coker}}\nolimits (P)$  
  $\displaystyle =$ $\displaystyle \dim\ker(P) - \dim\ker(P^*).$  

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

Properties

Fredholm operators of the form $ \mathit{invertible} + \mathit{compact}$ have index zero.



"Fredholm index" is owned by mhale.
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See Also: Fredholm operator

Other names:  index
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Cross-references: path, continuous, norm, compact operator, vector spaces, finite-dimensional, well defined, Fredholm operator

This is version 6 of Fredholm index, born on 2002-12-30, modified 2007-08-07.
Object id is 3863, canonical name is FredholmIndex.
Accessed 4785 times total.

Classification:
AMS MSC47A53 (Operator theory :: General theory of linear operators :: Fredholm operators; index theories)

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