Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distiguised computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \otimes e_{j+a}$.

Suppose that I have a bounded operator $A: l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z}) \to l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z}) $ which is invariant under these translations. I.e. $T_a A T_{-a} = A$. Further, $A$ has the property that $A \left( e_j \otimes e_k \right) = 0 $ for $\vert j-k \vert > R$ for some $R$. Now, analogusly with translation invariance in $ l^2(\mathbb{Z})$ I'd like to do some kind of Fourier-transformation to find a basis where the operator is nicer.

I suggest $\mathcal{F}: l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z}) \to \oplus_{r \in \mathbb{Z}} L^2 \left( \lbrack 0,2\pi \rbrack \right)_r$ defined by $(\mathcal{F}\psi)(k,r) = \sum_{x \in \mathbb{Z}} e^{-ikx} \psi(x,x-r)$ and inverse $(\mathcal{F}^{-1}\phi)(x,y) = \int_{0}^{2\pi} e^{ikx} \phi(k,x-y) dk $ with some normalization constants.

Does this make sense and if yes could you point me to some litterature about it? Further, if this does make sense how would one go about showing that $\mathcal{F}$ is unitary? I am ultimately interested in the spectrum of $A$.