is easy to see that $A_0$ is self-adjoint and that $$  \sigma(A_0) = \sigma_{\textup{ess}}(A_0) = (-\infty, -m c^2] \cup [m c^2, \infty),$$ cf. [@T92 Section 1.4]. Next, for $\lambda \in \rho(A_0) = \mathbb{C} \setminus \big( (-\infty, -m c^2] \cup [m c^2, \infty) \big)$ the resolvent of $A_0$ is $$(A_0 - \lambda)^{-1} f(x) = \int_{\mathbb{R}^3} G_\lambda(x-y) f(y) \textup{d} y, \quad f \in L^2(\mathbb{R}^3)^4,~x \in \mathbb{R}^3,$$ where the Green function $G_\lambda$ is given for $x\neq0$ by $$ \begin{split}
   G_\lambda(x) = \left( \frac{\lambda}{c^2} I_4 + m \beta 
                   + \left( 1 - i \sqrt{\frac{\lambda^2}{c^2} - (m c)^2} |x| \right) \frac{i}{c |x|^2} \alpha \cdot x \right)
                   \cdot \frac{e^{i \sqrt{\lambda^2/c^2 - (m c)^2} |x|}}{4 \pi |x|}&,
 \end{split}$$ see [@T92 Section 1.E]; in the last formula the convention ${\textup{Im}\,}\sqrt{\lambda^2/c^2 - (m c)^2} > 0$ is used.

Auxiliary integral operators
----------------------------

In this subsection we introduce several families of integral operators which are related to the Green function $G_\lambda$ and which will play a crucial role later in the study of Dirac operators with singular $\delta$-shell interactions. For a fixed $\lambda \in \rho(A_0) = \mathbb{C} \setminus \big( (-\infty, -m c^2] \cup [m c^2, \infty) \big)$ we define the potential operator $\Phi_\lambda: L^2(\Sigma)^4 \rightarrow L^2(\mathbb{R}^3)^4$ by $$\label{def_Phi_lambda}
  \Phi_\lambda \varphi(x) 