\right)
                   \cdot \int_{\Sigma} \frac{e^{i \sqrt{\lambda^2/c^2 - (m c)^2} |x-y|}}{4 \pi |x-y|} \varphi(y) \text{d} \sigma(y),$$ i.e. $\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta$ is a constant matrix times the single-layer boundary integral operator associated to $-\Delta + (m c)^2 - \frac{\lambda^2}{c^2}$, cf. [@M00 equation (9.15)]. This together with [@M00 Theorem 6.11], the fact that $H^{1/2}(\Sigma)^4$ is compactly embedded in $L^2(\Sigma)^4$, see, e.g., [@HW08 Theorem 4.2.2], and [@BH17 Proposition 4.4 (iii)], see also [@OV17 Proposition 2.8], yields the following proposition:

\[proposition\_mapping\_properties\] Let $\lambda \in \rho(A_0) = \mathbb{C} \setminus \big( (-\infty, - m c^2] \cup [m c^2, \infty) \big)$. Then the following holds.

-   The operator $\mathcal{C}_\lambda^2 - \frac{1}{4 c^2} I_4$ gives rise to a bounded operator $$\mathcal{C}_\lambda^2 - \frac{1}{4 c^2} I_4: L^2(\Sigma)^4 \rightarrow H^{1/2}(\Sigma)^4.$$ In particular, $\mathcal{C}_\lambda^2 - \frac{1}{4 c^2} I_4$ is compact in $L^2(\Sigma)^4$.

-   The operator $\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta$ gives rise to a bounded operator $$\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta: L^2(\Sigma)^4 \rightarrow H^{1/2}(\Sigma)^4.$$ In particular, $\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta$ is compact in $L^2(\Sigma)^4$.

Finally, we note that the adjoint $\Phi_\lambda^*: L^2(\mathbb{R}^3)^4 \rightarrow L^2(\Sigma)^4$ of $\Phi_\lambda$ acts as $$\label{equation_Phi_lambda_star}
  \Phi_\lambda^* f = \big( (A_0 - \overline{\lambda})^{-1} f \big)\big|_\Sigma$$ or, in a more explicit way, $$\Phi_\lambda^* f(x) = \int_{\mathbb{R}^3} G_{\bar{\lambda}}(x-y) f(y) 