$$\begin{aligned}
\Delta_{\nu,\mu} R_{\nu ,\mu}^{\lambda}f
=\lambda R_{\nu, \mu}^{\lambda}f-f, \  R_{\nu ,\mu}^{\lambda}f \in \mathcal{D}^*_{ \gamma}( \Delta_{\nu,\mu} ) .\end{aligned}$$ gives $$\begin{aligned}
\Delta_{\nu\circ F_{\mu}^{-1},\Lambda} (R_{\nu ,\mu}^{\lambda}f\circ \check{F}^{-1}_{\mu}) \circ {F}_{\mu}
=\lambda R_{\nu,\mu}^{\lambda}f-f\end{aligned}$$ and hence $$\begin{aligned}
\Delta_{\nu\circ F_{\mu}^{-1},\Lambda} (R_{\nu ,\mu}^{\lambda}f\circ \check{F}^{-1}_{\mu}) 
=\lambda R_{\nu,\mu}^{\lambda}f\circ \check{F}_{\mu}^{-1} -f \circ \check{F}_{\mu}^{-1}\end{aligned}$$ proving the first part. The statement on the norms now follows from .

\[Feller\] Let $\gamma \in[0,{\pi/2}]^{2}$. Then the operator $\Delta_{\mu,\nu}$ with domain $ {\mathcal{D}}_{\gamma}^s(\Delta_{\nu,\mu})$ is an infinitesimal generator of a strongly continuous semi-group of contraction on $C_{\nu,\mu}^{\gamma}$.

It is well known that this holds true for the classical Krein-Feller operator [@MR0314125 Behauptung 4.1]. This in tandem with the Yosida-Hille Theorem [@ma1992introduction p. 11, Theorem 1.12] and , yields $$\begin{aligned}
 \big\Vert R^{\lambda}_{\nu \circ F_{\mu}^{-1},\Lambda} \big\Vert_{C^{\gamma}_{\nu \circ F_{\mu}^{-1},\Lambda}}
=
 \big\Vert R^{\lambda}_{\nu,\mu} \big\Vert_{C_{\nu,\mu}^{\gamma}} \leq 1/\lambda ,\end{aligned}$$ for all $\lambda>0$. Further $\Delta_{\nu, \mu}$ is densely defined and closed operator in $C_{\nu,\mu}^{\gamma}$. Applying again the theorem of Yosida-Hille gives the statement.

Applications
============

Spectral asymptotics
--------------------

In this section we review the asymptotic spectral properties of $\Delta_{\nu,\mu}$ for certain classes of self-similar measures. We will show how the results in [@Fr05] can be deduced from [@Fu87] with the help of the above established isomorphism. Let us give general assumption on the self-similar measures 