I am reading the 2006 book Spectra and Pseudospectra by Trefethen. On pages 50-51, it is stated that for the following Laurent series
$$f(z)=\sum_k a_k z^k=2z^{-3}-z^{-2}+2i z^{-1}-4z^2-2iz^3$$
and
$$A=\begin{pmatrix} 0 & 2i & -1 & 2 & 0 & 0 \newline 0 & 0& 2i & -1 & 2 & 0 \newline -4 & 0 & 0 & 2i & -1 & 2 \newline -2i & -4 & 0 & 0 & 2i & -1 \newline 0 & -2i & -4 & 0 & 0 & 2i \newline 0 & 0 & -2i & -4 & 0 & 0\end{pmatrix}$$
then the Laurent operator $A$ defined by a vector $a=(a_j)$ is equivalent to a convolution $Au=a \ast u$.
I have spent quite some time and still don't understand what it means. Can someone briefly explain the relationship between Laurent series $f(z)$ and Laurent/Toeplitz operator?
By definition a "symbol" $f$ which is $2\pi$-periodic ($L^1(-\pi,\pi)$) generates a Toeplitz matrix $T_n(f)$ by
$$T_n(f)=\begin{bmatrix}f_0&f_{-1}&\ldots& &f_{1-n}\\f_1&f_0&f_{-1}\\\vdots&\ddots&\ddots&\ddots\\ &&f_1&f_0&f_{-1}\\ f_{n-1}&&&f_1&f_0\end{bmatrix}$$ where $f_k$ are the Fourier coefficients fo $f$. In your case set $z=e^{i\theta}$, i.e. parametrize on the unit circle.
$$f(\theta)=2e^{-3i\theta}-e^{-2i\theta}+2e^{-i\theta}-4e^{2i\theta}-2ie^{3i\theta}$$
As a side note, if $f$ is real-valued, then $T_n(f)$ is Hermitian. An approximation of the singular values of $T_n(f)$ is to sample the modulus of the symbol with an equispaced grid $\theta_j\in[-\pi,\pi]$, $\sigma_j(T_n(f))\approx|f(\theta_j)|$. If $f$ is Hermitian you can approximate the eigenvalues as $\lambda_j(T_n(f))\approx f(\theta_j)$. Look at the Szegő limit theorems. You can study this in the books by Böttcher, Silbermann, Grudsky, etc.