:= \frac{1}{\sqrt{\pi}} \sum_{j = 0}^\infty
\chi_{n + j} \sin \big( (j + 1) \theta \big).$$ Notice that the inner product in $L^2_{\rm odd} (\mathbb{T},Q_{n,n})$ is defined by $$\label{eq3,12}
\langle f,g \rangle_{L^2_{\rm odd}} := \int_{\mathbb{T}} \langle f(\theta),g(\theta) 
\rangle \, d\theta.$$ Hence, a direct computation shows that $$\label{eq3,13}
\langle U \chi,U \chi \rangle_{L^2_{\rm odd}} = 
\langle \chi,\chi \rangle_{\ell^2({{\mathbb{Z}}}^+,Q_{n,n})} = \langle \phi,\phi \rangle,$$ where the last equality corresponds to . Moreover, we have the following lemma:

[[@af Lemma 2]]{}\[l3,2\] For any $n \geq 0$, we have $$\label{eq3,14}
U L_n U^\ast = 2\sqrt{k} \cos (\theta).$$

In particular, Lemma \[l3,2\] shows that for any $n \geq 0$, the spectrum of the operator $L_n$ is equal to $\big[ -2\sqrt{k},2\sqrt{k} \big]$ and is absolutely continuous. Hence, this implies .

Representation of the weighted resolvent $A (-L + k + 1 - z)^{-1} B^\ast$ {#sec4}
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In this section, we give a suitable representation of the weighted resolvent $A (-L + k + 1 - z)^{-1} B^\ast$ which turns to be useful in our analysis, where $A$ and $B$ are bounded operators on $\ell^2(\mathcal{V})$.

For $n \geq 0$, let $P_n$ be the orthogonal projection of $\ell^2(\mathcal{V})$ onto $M_n$, the subspace defined in Lemma \[l3,1\]. Since $M_n$ can be 