 Laurent operator

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 $\ell (\theta )$, andThe 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, $\ell (\theta )={\sum }_j{a}_j{e}^{ij\theta }$. The neighbor-sum operator $L{u}_m=u_{m-1}+u_{m+1}$ 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.