PlanetMath: Fredholm index

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Fredholm index

(Definition)

Let be a Fredholm operator. The index of 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

Fredholm.

Properties

$\mathop{\mathrm{index}}\nolimits (P^*) = -\mathop{\mathrm{index}}\nolimits (P)$ .
$\mathop{\mathrm{index}}\nolimits (P+K) = \mathop{\mathrm{index}}\nolimits (P)$ for any compact operator .
If $P_1\colon \mathord{\mathcal{H}}_1 \to \mathord{\mathcal{H}}_2$ and $P_2\colon \mathord{\mathcal{H}}_2 \to \mathord{\mathcal{H}}_3$ are Fredholm operators, then $\mathop{\mathrm{index}}\nolimits (P_2 P_1) = \mathop{\mathrm{index}}\nolimits (P_1) + \mathop{\mathrm{index}}\nolimits (P_2)$ .
If $t \to P_t$ , $t \in [0,1]$ is a norm continuous path of Fredholm operators, then $\mathop{\mathrm{index}}\nolimits (P_t) = \mathop{\mathrm{index}}\nolimits (P_0)$ .

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

"Fredholm index" is owned by mhale.

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