proof of Theorem \[main\] below. The first of them has been used, e.g., in [@Pejsachowicz §7].

\[lemma-sflperturbation\] Let $\mathcal{A}:I\rightarrow\mathcal{CF}^\textup{sa}(H)$ be gap-continuous and set $\mathcal{A}^\delta=\mathcal{A}+\delta I_H$ for $\delta\in\mathbb{R}$. Then, for any sufficiently small $\delta>0$, $\mathcal{A}^\delta$ is a gap-continuous path in $\mathcal{CF}^\textup{sa}(H)$ and

$$\begin{aligned}
\label{sfldelta}
\operatorname{sf}(\mathcal{A})=\operatorname{sf}(\mathcal{A}^\delta).\end{aligned}$$

We note at first that the operators $\mathcal{A}^\delta_\lambda$ are selfadjoint and Fredholm for $\delta$ sufficiently small, which follows from standard stability theory (see e.g. [@Kato]). Moreover, the path $\mathcal{A}^\delta$ is gap-continuous by [@Kato Thm. IV.2.17], and so $\operatorname{sf}(\mathcal{A}^\delta)$ is well defined.\
To show , let $0=\lambda_0<\ldots<\lambda_N=1$ be a partition of the unit interval and $\varepsilon_i>0$, $i=1,\ldots,N$, for $\mathcal{A}$ as in . Let $\mathcal{N}_{T,\varepsilon_i}$ be an open neighbourhood of some $T\in\mathcal{CF}^{sa}(H)$ as in the construction of the spectral flow such that $\mathcal{A}_\lambda\in\mathcal{N}_{T,\varepsilon_i}$ for all $\lambda\in[\lambda_{i-1},\lambda_i]$. Now there is $\delta_i>0$ such that $\mathcal{A}^{s\delta_i}_\lambda\in\mathcal{N}_{T,\varepsilon_i}$ for all $s\in[0,1]$ and all $\lambda\in [\lambda_{i-1},\lambda_i]$, i.e. the spectral projections $\chi_{[-\varepsilon_i,\varepsilon_i]}(\mathcal{A}^{s\delta_i}_{\lambda})$ are of the same finite rank. Moreover, by choosing $\delta_i>0$ smaller, we can assume that

$$\sigma(\mathcal{A}_{\lambda_i})\cap[-\delta_i,0)=\sigma(\mathcal{A}_{\lambda_{i-1}})\cap[-\delta_i,0)=\{0\}.$$ Then, as $\sigma(\mathcal{A}^{\delta_i}_\lambda)=\sigma(\mathcal{A}_\lambda)+\delta_i$, we see that

$$\dim(\operatorname{im}(\chi_{[0,\varepsilon_i]}(\mathcal{A}_{\lambda})))=\dim(\operatorname{im}(\chi_{[0,\varepsilon_i]}(\mathcal{A}^{\delta_i}_{\lambda}))),\quad \lambda=\lambda_{i-1},\lambda_i.$$ If we now set $\delta=\min\{\delta_1,\ldots,\delta_N\}>0$, then $$\dim(\operatorname{im}(\chi_{[0,\varepsilon_i]}(\mathcal{A}_{\lambda})))=\dim(\operatorname{im}(\chi_{[0,\varepsilon_i]}(\mathcal{A}^{\delta}_{\lambda}))),\quad \lambda=\lambda_{i-1},\lambda_i$$ holds simultaneously for this $\delta$ and all $i=1,\ldots,N$, and so the assertion follows from the definition .

Finally, let us note 