density $\varphi \in H^{1/2}(\Sigma)^4$ such that $$\label{equation_kernel}
  f_\lambda = \Phi_\lambda \varphi;$$ see [@BH17 Proposition 4.2].

Now, we describe how $\Phi_\lambda$ and $\mathcal{C}_\lambda$ are related to each other by taking traces. Let $\varphi \in H^{1/2}(\Sigma)^4$ and $\lambda \in \rho(A_0)$. Then, the trace of the function $$\Phi_\lambda \varphi = (\Phi_\lambda \varphi)_+ \oplus (\Phi_\lambda \varphi)_-  
  \in H^1(\Omega_+)^4 \oplus H^1(\Omega_-)^4$$ on $\Sigma$ is $$\big((\Phi_\lambda \varphi)_\pm\big) \big|_\Sigma = \mathcal{C}_\lambda \varphi \mp \dfrac{i}{2 c} (\alpha \cdot \nu) \varphi;$$ this is shown in [@AMV15 Lemma 2.2] for $\lambda \in (-m c^2, m c^2)$, the case $\lambda \in \mathbb{C} \setminus \mathbb{R}$ can be proved in the same way. In particular, using $(\alpha \cdot \nu)^2 = I_4$ one finds that the identities $$\begin{gathered}
 \label{jump1}
  \frac{1}{2} \big( (\Phi_\lambda \varphi)_+|_\Sigma + (\Phi_\lambda \varphi)_-|_\Sigma \big) = \mathcal{C}_\lambda \varphi,\\
 \label{jump2}
  i c \alpha \cdot \nu \big( (\Phi_\lambda \varphi)_+|_\Sigma - (\Phi_\lambda \varphi)_-|_\Sigma \big) = \varphi,\end{gathered}$$ hold. Finally, let us mention the mapping properties of the operators $\mathcal{C}_\lambda^2 - \frac{1}{4 c^2} I_4$ and $\mathcal{C}_\lambda \beta + \beta \mathcal{C}_\lambda$ which will be important for the analysis of $A_{\eta, \tau}$. Using the anti-commutation relation  it is easy to see for $\varphi \in L^2(\Sigma)^4$ that $$(\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta) \varphi(x) =  2 \left(\frac{\lambda}{c^2} \beta + m I_4 