g  \right\rangle_{L^2(\nu)}
=\left\langle f\circ \check{F}^{-1}_{\mu} ,  \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda} (g \circ \check{F}^{-1}_{\mu})   \right\rangle_{L^2(\nu \circ F_{\mu}^{-1})}
 \end{aligned}$$ defines a continuous linear functional on $ \operatorname{dom}(\Delta_{\nu,\mu})$. Combining Proposition \[adjoint\] and Theorem \[prop:laplace\_backward\] we deduce $ f  \circ \check{F}^{-1}_{\mu} \in \operatorname{dom}\big(  \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda}^{*}\big)= \operatorname{dom}\big(  \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda}\big)={\mathcal{D}}_{\gamma}^w( \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda})$ and consequently $f  \in {\mathcal{D}}_{\gamma}^w(\Delta_{\nu,\mu})=\operatorname{dom}(\Delta_{\nu,\mu})$.

There is an analogous theorem in general theory of Markov processes \[[@DynkinI_II], p. 325, Theorem 10.13\]. Taking into account that $\Delta_{\nu,\mu}$ with domain ${\mathcal{D}}_{\gamma}^{*}(\Delta_{\nu,\mu})$ has a probabilistic interpretation as infinitesimal generator of a Markov process, thus the result above is not surprising.

Corollary \[Spectral\] implies that $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$ with domain ${\mathcal{D}}_{\gamma}^w \left( \Delta_{\nu, \Lambda} \right)$ gives rise to an orthonormal basis of eigenfunctions with non-positive eigenvalues. In the case of ${\operatorname{supp}}(\nu)$ is infinite we have $(\lambda_n)_{n \in \mathbb{N}}$ with $\lim_{n \to \infty}-\lambda_n = \infty$ otherwise there are only finitely many eigenvalues. Using the one-to-one correspondence established in to relate the spectral properties of $\Delta_{\nu,\mu}$ with those of $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$, we obtain the following.

\[thm Spec\] For fixed $ \gamma \in [0,{\pi/2}]^{2}$, the operators $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} $ with domain $\mathcal{D}^w_{\gamma}( \Delta_{\nu \circ F_{\mu}^{-1},\Lambda} )$ and $ \Delta_{\nu,\mu}  $ with 