+ (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big)^{-1}: L^2(\Sigma)^4 \rightarrow L^2(\Sigma)^4$$ exists and is bounded and everywhere defined.

\(i) If $\big(I_4 + (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big) \varphi$ belongs to $H^{1/2}(\Sigma)^4$, then by  also $$    \begin{split}
      \psi &:= \big( I_4 - \mathcal{C}_\lambda (\eta I_4 - \tau \beta)  \big) \big(I_4 + (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big) \varphi \\
      &= \left( 1 - \frac{\eta^2 - \tau^2}{4 c^2} \right) \varphi + \tau (\mathcal{C}_\lambda \beta + \beta \mathcal{C}_\lambda) \varphi + (\eta^2 - \tau^2) \left( \frac{1}{4 c^2} I_4 - \mathcal{C}_\lambda^2 \right) \varphi
    \end{split}$$ belongs to $H^{1/2}(\Sigma)^4$. Making use of Proposition \[proposition\_mapping\_properties\] this implies that also $$\varphi = \frac{4 c^2}{4 c^2 - \eta^2 + \tau^2} \left( \psi - \tau (\mathcal{C}_\lambda \beta + \beta \mathcal{C}_\lambda) \varphi - (\eta^2 - \tau^2) \left( \frac{1}{4 c^2} I_4 - \mathcal{C}_\lambda^2 \right) \varphi \right)$$ belongs to $H^{1/2}(\Sigma)^4$, which is the claim of item (i).

\(ii) Assume first that $\lambda \in \rho(A_0)$ is an eigenvalue of $A_{\eta, \tau}$ with eigenfunction $f_\lambda \neq 0$. Then, according to  there exists a density $0 \neq \varphi \in H^{1/2}(\Sigma)^4$ such that $f_\lambda = \Phi_\lambda \varphi$. Since $f_\lambda \in {\mathrm{dom}\,}A_{\eta, \tau}$ this function fulfils . Using  and this yields $$\label{equation_Birman_Schwinger}
    \begin{split}
      0       &= i 