In this artice Spectral pollution and eigenvalue bounds by Lyonell Boulton he has at page 7
Let $S$ be the multiplication operator by the "sign" function, $S f(t)=\operatorname{sign}(t) f(t) \quad S: L^2(-\pi, \pi) \longrightarrow L^2(-\pi, \pi)$. Then $$ \operatorname{spec} S=\operatorname{spec}_{\mathrm{ess}} S=\{ \pm 1\} . $$ The discrete Fourier transform $\mathcal{U}: L^2(-\pi, \pi) \longrightarrow \ell^2(\mathbb{Z})$ $$ \mathcal{U} f=(\hat{f}(n))_{n \in \mathbb{Z}} \quad \hat{f}(n)=\frac{1}{\sqrt{2 \pi}} \int_{-\pi}^\pi f(t) e^{-i n t} \mathrm{~d} t $$ is an invertible isometry. The corresponding Laurent operator associated to $S$ is $$ \left.L(\operatorname{sign})=\mathcal{U S U}^{-1}=\widehat{\operatorname{sign}}(j-k)\right]_{j k=-\infty}^{\infty}: \ell^2(\mathbb{Z}) \longrightarrow \ell^2(\mathbb{Z}) . $$ Its spectrum coincides exactly with that of $S$.
I am not understanding what are $j,k$ in the operator $$ \left.L(\operatorname{sign})=\mathcal{U S U}^{-1}=\widehat{\operatorname{sign}}(j-k)\right]_{j k=-\infty}^{\infty}: \ell^2(\mathbb{Z}) \longrightarrow \ell^2(\mathbb{Z}) . $$
The operator $\mathcal U\mathcal S\mathcal U^{-1}$ should be a convolution operator, since the Fourier transform maps products in convolutions and viceversa. Thus, the operator $\mathcal{U}\mathcal{S}\mathcal{U}^{-1}$ is the operator that maps a sequence $(a(n))_{n\in\mathbb Z}\in\ell^2(\mathbb Z)$ into the sequence $(\mathcal{U}\mathcal{S}\mathcal{U}^{-1} a)$ defined as $$ (\mathcal{U}\mathcal{S}\mathcal{U}^{-1} a)(n)=\sum_{k\in\mathbb Z}\widehat {\operatorname{sign}}(n-k)a(k), $$ where $\widehat {\operatorname{sign}}=\mathcal U( {\operatorname{sign}}) $.
The notation is similar to that of a matrix: to obtain $\mathcal{U}\mathcal{S}\mathcal{U}^{-1} a$, that is nothing but a vector with infinite components, you take the matrix product between the infinite matrix $(\widehat {\operatorname{sign}}(j-i))_{i,j\in\mathbb Z}$ and the vector $(a(n))_{n\in\mathbb Z}$, which formally is the same thing as doing the convolution between $a$ and $\widehat {\operatorname{sign}}$. So, $j$ and $k$ in your question are the indices of the matrix that represents the operator $\mathcal{U}\mathcal{S}\mathcal{U}^{-1}$ in the standard basis of $\ell^2(\mathbb Z)$.