M_n$, i.e. $$\label{eq3,7}
\phi = \displaystyle\oplus_{j = 0}^{\infty} \phi_{n,n+j},$$ with $\phi_{n,n+j} \in Q_{n,n+j}$ for any $j \geq 0$. The idea is to construct an isomorphism between the subspace $M_n$ and the space of $Q_{n,n}$-valued sequences, namely the space $\ell^2({{\mathbb{Z}}}^+,Q_{n,n})$. By construction of the $Q_{n,r}$, $n \geq 0$, $r \geq 0$ (see for instance [@af]), for any $j \geq 0$, the operator $\big( \frac{1}{\sqrt{k}} \Pi \big)^j$ defines an isometry between $Q_{n,n+j}$ and $Q_{n,n}$, and can be written as $$\label{eq3,8}
\phi = \oplus_{j = 0}^{\infty} \left( \frac{1}{\sqrt{k}} \Pi \right)^j \chi_{n + j},$$ where $\chi := (\chi_{n + j})_{j \geq 0}$ defines a sequence of vectors lying in $Q_{n,n}$. Therefore, under the above considerations, the operator $$\label{eq3,9}
W : M_n \longrightarrow \ell^2({{\mathbb{Z}}}^+,Q_{n,n}), \quad W \phi = \chi,$$ defines an isomorphism between the spaces $M_n$ and $\ell^2({{\mathbb{Z}}}^+,Q_{n,n})$. Indeed, we have $$\label{eq3,90}
\langle \phi,\phi \rangle = \sum_{j = 0}^{\infty} \langle \phi_{n + j},\phi_{n + j} 
\rangle = \sum_{j = 0}^{\infty} \langle \chi_{n + j},\chi_{n + j} 
\rangle = \langle \chi,\chi \rangle_{\ell^2({{\mathbb{Z}}}^+,Q_{n,n})}.$$ Now, let ${{\mathbb{T}}}:= {{\mathbb{R}}}/2\pi {{\mathbb{Z}}}$ be the torus and define the unitary operator $$\label{eq3,10}
U : M_n \cong \ell^2({{\mathbb{Z}}}^+,Q_{n,n}) \longrightarrow L^2_{\rm odd} (\mathbb{T},Q_{n,n})$$ acting as $$\label{eq3,11}
U \big( (\chi_n,\chi_{n + 1}, \ldots ) \big) 