we set $H=\S_{\gamma}^w$, $A= \Delta_{\nu, \mu}$ and $\operatorname{dom}(A)={\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )\cap \S_{\gamma}^w$. In the case of $\gamma \in [0, \pi/2] \setminus \{ (\pi/2,\pi/2)\}$ it follows immediately that $ \Delta_{\nu, \mu}$ with domain ${\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )$ is a densely defined, self-adjoint operator on $L^{2}(\nu)$. In the case of Neumann boundary condition, i.e. $\gamma=(\pi/2,\pi/2)$, note that for the kernel we have $\Delta_{\nu,\ \mu}^{-1}(\{0\})=  \R\mathds{1}$ and hence $L^2(\nu) = \S_{\gamma}^w \oplus   \R\mathds{1}$. Therefore we have $ {\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )=\{f +a:f\in {\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )\cap \S_{\gamma}^w, a \in \R\}$ is dense in $L^2(\nu) $. Using this oberservation it follows again that $\Delta_{\nu, \mu} $ with domain ${\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )$ is a densely defined and self-adjoint operator on $L^2(\nu) $. The following proposition summarizes this observation.

\[adjoint\] The partially defined operator $\Delta_{\nu,\Lambda}:L^{2}(\nu)\to L^{2}(\nu) $ with domain $ \mathcal{D}^{w}_{\gamma}( \Delta_{\nu,\Lambda}) $ for $\gamma  \in [0,{\pi/2}]^{2}$ is self-adjoint, non-positive and, in particular, closed.

Fix $\gamma=(\alpha,\beta) \in [0,\pi/2]^2\setminus\{ (\pi/2, \pi/2)\}$ then the operator $R_0\coloneqq  -\Delta_{\nu,\Lambda}^{-1}: \S_{\gamma}^w \to \S_{\gamma}^w$ is compact and self-adjoint.

Lemma \[Surjec\] shows that $R_0$ is a Hilbert-Schmidt operator with continuous (bounded) kernel function, therefore the compactness follows. Furher, the operator $R_0$ is self-adjoint this follows from symmetry of $\Delta_{\nu, \Lambda}$ in tandem with the fact that $R_0$ 