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Fredholm index
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(Definition)
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Let $P$ be a Fredholm operator. The index of $P$ is defined as \begin{eqnarray*} \ind(P) & = & \dim\ker(P) - \dim\coker(P) \\ & = & \dim\ker(P) - \dim\ker(P^*). \end{eqnarray*}Note: this is well defined as $\ker(P)$ and $\ker(P^*)$ are finite-dimensional vector spaces, for $P$ Fredholm.
Properties
- $\ind(P^*) = -\ind(P)$ .
- $\ind(P+K) = \ind(P)$ for any compact operator $K$ .
- If $P_1\colon \hilbert_1 \to \hilbert_2$ and $P_2\colon \hilbert_2 \to \hilbert_3$ are Fredholm operators, then $\ind(P_2 P_1) = \ind(P_1) + \ind(P_2)$ .
- If $t \to P_t$ , $t \in [0,1]$ is a norm continuous path of Fredholm operators, then $\ind(P_t) = \ind(P_0)$ .
Fredholm operators of the form $\mathit{invertible} + \mathit{compact}$ have index zero.
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"Fredholm index" is owned by mhale.
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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 6157 times total.
Classification:
| AMS MSC: | 47A53 (Operator theory :: General theory of linear operators :: Fredholm operators; index theories) |
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Pending Errata and Addenda
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