+ k + 1 - z)^{-1} B^\ast 
\varphi,\psi \biggr\rangle \\
& = \sum_{n \, \geq \, 0} \biggl\langle U (-L_n + k + 1 - z)^{-1} U^\ast U P_n
B^\ast \varphi,U P_n A^\ast \psi 
\biggr\rangle_{L^2_{\rm odd}},
\end{split}$$ $\langle \cdot,\cdot \rangle_{L^2_{\rm odd}}$ being the inner product defined by . Together with Lemma \[l3,2\], this gives $$\label{eq4,9}
\begin{split}
& \biggl\langle A (-L + k + 1 - z)^{-1} B^\ast
\varphi,\psi \biggr\rangle \\
& = \sum_{n \, \geq \, 0} \int_{\mathbb{T}} \big( -2\sqrt{k} \cos (\theta) + k + 1 - z \big)^{-1} 
\biggl\langle U P_n B^\ast \varphi (\theta),U P_n 
A^\ast \psi (\theta) \biggr\rangle \, d\theta.
\end{split}$$ From , it follows that for any $\phi \in \ell^2(\mathcal{V})$ and any bounded operator $W$ on $\ell^2(\mathcal{V})$, we have $$\label{eq4,10}
U P_n W \phi (\theta) = \sum_{j \, \geq \, 0} 
\sum_{0 \leq m \leq N_j} \biggl\langle \phi,W^\ast E_m^{n,n+j} 
\biggr\rangle U E_m^{n,n+j} (\theta).$$ Then, combining and , we obtain $$\label{eq4,11}
\begin{split}
 \biggl\langle & A (-L + k + 1 - z)^{-1} B^\ast
\varphi,\psi \biggr\rangle \\
& = \sum_{n \, \geq \, 0} \sum_{\substack{j \, \geq \, 0 \\ \ell \, \geq \, 0}} 
\sum_{\substack{0 \leq m \leq N_j \\ 0 \leq q \leq N_\ell}} 
\biggl\langle \varphi,B E_m^{n,n+j} 
