Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished 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, analogously 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 literature 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 $A$. 



Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$
Fine question. The first thing to notice is that your problem becomes simpler if you work in the $(1,1)$, $(0,1)$ basis of $\mathbb{Z}\times\mathbb{Z}$. I.e., we have an isomorphism from $\mathbb{Z}\times \mathbb{Z}$ to itself given by the matrix $ \left[\begin{matrix}1&0\cr
1&1
\end{matrix}\right]$, and conjugating by this turns your operator into one which commutes with horizontal translations.


Then we can take the Fourier transform in the first variable to go to $L^2(\mathbb{T})\otimes l^2(\mathbb{Z})$ --- these two steps, change basis and Fourier transform are effectively what you have done. (BTW you can see directly that it's unitary by noticing that it takes the orthonormal basis $(e\_i\otimes e\_j)$ to an orthonormal basis of the range.) You now have an operator on this Hilbert space which commutes with multiplication by the first variable, which means it commutes with all of the spectral projections of that multiplication operator, which means the subspace $L^2(S)\otimes l^2(\mathbb{Z})$ is invariant, for any measurable $S \subseteq \mathbb{T}$.


Actually, the band-limiting condition means it's effectively an operator from $L^2(\mathbb{T})\otimes l^2[-R,R]$ to $L^2(\mathbb{T})\otimes l^2(\mathbb{Z})$.


I can't say how that will help with your problem without knowing more about $A$, but I think it cleanly packages the information you've given us. I guess you can write $A$ as a direct integral over $\mathbb{T}$ of a family of operators on $l^2(\mathbb{Z})$, and you can relate the spectrum of $A$ to the spectra of these operators, but I somehow doubt that would really help. I can give references for that last comment if you like, the rest is just standard functional analysis.


(Actually, a better way to say this is that after the transformation your operator belongs to $B(l^2(\mathbb{Z}))\otimes L^\infty(\mathbb{T})$.)


