the following stability of the spectral flow under conjugation by invertible operators, where we denote by $M^T$ the adjoint of an operator in the real Hilbert space $H$.

\[lemma-conjuagtion\] Let $\mathcal{A}:I\rightarrow\mathcal{CF}^\textup{sa}(H)$ be a gap-continuous path and $M:I\rightarrow GL(H)$ a continuous family of bounded invertible operators. Then $\{M^T_\lambda\mathcal{A}_\lambda M_\lambda\}_{\lambda\in I}$ is gap-continuous and

$$\operatorname{sf}(M^T\mathcal{A}M)=\operatorname{sf}(\mathcal{A}).$$

Note that

$$\begin{aligned}
\operatorname{graph}(M^T_\lambda\mathcal{A}_\lambda M_\lambda)&=\{(u,M^T_\lambda\mathcal{A}_\lambda M_\lambda u):\, u\in M^{-1}_\lambda(\mathcal{D}(\mathcal{A}_\lambda))\}=\{(M^{-1}_\lambda v,M^T_\lambda\mathcal{A}_\lambda v):\,v\in\mathcal{D}(\mathcal{A}_\lambda)\}\\
&=\begin{pmatrix}
M^{-1}_\lambda&0\\
0&M^T_\lambda
\end{pmatrix}\,\operatorname{graph}(\mathcal{A}_\lambda)=:N_\lambda\operatorname{graph}(\mathcal{A}_\lambda)\subset H\times H,\end{aligned}$$

and so $\{N_\lambda P_{\operatorname{graph}(\mathcal{A}_\lambda)}N^{-1}_\lambda\}_{\lambda\in I}$ is a continuous family of oblique projections onto$\{\operatorname{graph}(M^T_\lambda\mathcal{A}_\lambda M_\lambda)\}_{\lambda\in I}$ in $\mathcal{L}(H\times H)$. By [@Kato Thm. I.6.35], we have for the corresponding orthogonal projections $P_{\operatorname{graph}(M^T_\lambda\mathcal{A}_\lambda M_\lambda)}$ onto $\operatorname{graph}(M^T_\lambda\mathcal{A}_\lambda M_\lambda)$ the inequality

$$\|P_{\operatorname{graph}(M^T_\mu\mathcal{A}_\mu M_\mu)}-P_{\operatorname{graph}(M^T_\lambda\mathcal{A}_\lambda M_\lambda)}\|\leq \|N_\mu P_{\operatorname{graph}(\mathcal{A}_\mu)}N^{-1}_\mu-N_\lambda P_{\operatorname{graph}(\mathcal{A}_\lambda)}N^{-1}_\lambda\|,\quad \mu,\lambda\in I.$$ Consequently, $\{P_{\operatorname{graph}(M^T_\lambda\mathcal{A}_\lambda M_\lambda)}\}_{\lambda\in I}$ is continuous, which shows that $M^T\mathcal{A}M$ is gap-continuous.\
For the equality of the spectral flows, we just need to note that $M$ is homotopic inside $GL(H)$ to the constant path given by the identity $I_H$. Let us point out that this does not even require Kuiper’s Theorem as we just need to shrink $M$ to a constant path and use that $GL(H)$ is connected. As the conjugation preserves kernel dimensions, we obtain by the homotopy invariance (i) from above

$$\begin{aligned}
\operatorname{sf}(M^T\mathcal{A}M)=\operatorname{sf}(\mathcal{A}).\end{aligned}$$

Proof 