decomposed as $$\label{eq4,1}
M_n = \bigoplus_{j = 0}^{\infty} Q_{n,n + j},$$ then if we let $(E_m^{n,n+j})_{0 \leq m \leq N_j}$ denote an orthonormal basis of the finite-dimensional space $Q_{n,n + j}$ for any $j$ fixed, we have $$\label{eq4,2}
P_n = \sum_{j \, \geq \, 0} \sum_{0 \leq m \leq N_j} \langle \cdot,E_m^{n,n+j} \rangle 
E_m^{n,n+j}.$$ Notice that for any $j \geq 0$ fixed, we have $$\label{eq4,3}
1 + N_j = \dim \, Q_{n,n+j} < k^{n + j} = \dim \, \ell^2(S_{n + j}),$$ since $Q_{n,n+j} \subset \ell^2(S_{n + j})$. Furthermore, according to Section \[sec3\], for any $0 \leq m \leq N_j$ fixed, there exists a unique vector $\chi_m^{n,n+j} \in Q_{n,n}$ such that $$\label{eq4,4}
E_m^{n,n+j} = \left( \frac{1}{\sqrt{k}} \Pi \right)^j \chi_m^{n,n + j}.$$ Our goal in this section is to prove the following result:

\[p4,1\] Let $A$ and $B$ be two bounded operators on $\ell^2(\mathcal{V})$. Then, for any $z$ in the resolvent set of the operator $-L + k + 1$ and any $\varphi \in \ell^2(\mathcal{V})$, we have $$\label{eq4,5}
A (-L + k + 1 - z)^{-1} B^\ast 
\varphi (v) = \sum_{v' \in \mathcal{V}} K(v,v') \varphi (v'),$$ where the kernel $K(v,v')$ is given by $$\label{eq4,6}
\begin{split}
K(v,v') := & \frac{1}{2 \sqrt{k}} \sqrt{\frac{2}{\pi}} 