= Laurent operator
{wiki}
A bounded scalar <Laurent operator> on $\ell^2(\mathbb Z)$ commutes with the bilateral lattice shift. Its bi-infinite <matrix> has constant diagonals, formally $(Lu)_m=\sum_j a_j u_{m+j}$. Under the <discrete Fourier transform>, it becomes multiplication by an essentially bounded symbol $\ell(\theta)$, and
$$
\|L\|_{2\to2}=\operatorname*{ess\,sup}_\theta|\ell(\theta)|.
$$
The boundedness assumption is essential; a formal constant-diagonal matrix need not define a bounded map on $\ell^2$.
For a <finite difference operator> with finitely many nonzero coefficients, $\ell(\theta)=\sum_j a_je^{ij\theta}$. The neighbor-sum operator $Lu_m=u_{m-1}+u_{m+1}$ has symbol $2\cos\theta$ and <operator norm> two. Compression to a one-sided sequence space yields a <Toeplitz operator>, for which the boundary prevents the same full-lattice diagonalization.
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