It was shown in [@UnbSpecFlow] that for every $T\in\mathcal{CF}^\textup{sa}(H)$ there is $\varepsilon>0$ and a neighbourhood $\mathcal{N}_{T,\varepsilon}\subset\mathcal{CF}^\textup{sa}(H)$ of $T$ such that $\pm\varepsilon\notin\sigma(S)$ and the spectral projection $\chi_{[-\varepsilon,\varepsilon]}(S)$ is of finite rank for all $S\in\mathcal{N}_{T,\varepsilon}$. Let us now consider a path $\mathcal{A}=\{\mathcal{A}_\lambda\}_{\lambda\in I}$ in $\mathcal{CF}^\textup{sa}(H)$. There are $0=\lambda_0<\lambda_1<\ldots<\lambda_N=1$ such that the restriction of the path $\mathcal{A}$ to $[\lambda_{i-1},\lambda_i]$ is entirely contained in a neighbourhood $\mathcal{N}_{T_i,\varepsilon_i}$ as above for some $T_i\in\mathcal{CF}^\textup{sa}(H)$ and some $\varepsilon_i>0$. The *spectral flow* of the path $\mathcal{A}$ is defined as

$$\begin{aligned}
\label{sfl-def}
\operatorname{sf}(\mathcal{A})=\sum^N_{i=1}{\left(\dim(\operatorname{im}(\chi_{[0,\varepsilon_i]}(\mathcal{A}_{\lambda_i}))-\dim(\operatorname{im}(\chi_{[0,\varepsilon_i]}(\mathcal{A}_{\lambda_{i-1}}))\right)}.\end{aligned}$$

It follows by an argument of Phillips [@Phillips] that $\operatorname{sf}(\mathcal{A})$ only depends on the path $\mathcal{A}$, and that the following fundamental property holds (see also [@UnbSpecFlow]).

1.  Let $h:I\times I\rightarrow\mathcal{CF}^\textup{sa}(H)$ be a homotopy such that the dimensions of the kernels of $h(s,0)$ and $h(s,1)$ are constant for all $s\in I$. Then $$\operatorname{sf}(h(0,\cdot))=\operatorname{sf}(h(1,\cdot)).$$

Moreover, it is easily seen from the definition of the spectral flow that

1.  if the dimension of the kernel of $\mathcal{A}_\lambda$ is constant for all $\lambda\in I$, then $\operatorname{sf}(\mathcal{A})=0$;

2.  if $\mathcal{A}^1$ and $\mathcal{A}^2$ are two paths in $\mathcal{CF}^\textup{sa}(H)$ such that $\mathcal{A}^1_1=\mathcal{A}^2_0$, then

    $$\operatorname{sf}(\mathcal{A}^1\ast\mathcal{A}^2)=\operatorname{sf}(\mathcal{A}^1)+\operatorname{sf}(\mathcal{A}^2).$$

Let us finally note two further elementary properties of the spectral flow which play a crucial role in our 