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.