Monomial Rota-Baxter and averaging operators on $F[x,x^{-1}]$

The study of Rota-Baxter operators on the polynomial algebra in one variable was initiated by S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial Rota-Baxter operators on $F[x]$ were classified in 2016, while monomial averaging operators on $F[x]$ and $F_0[x]$ were classified in 2024. In the current work, we continue the study of these operators on Laurent polynomial algebra $F[x,x^{-1}]$ over a field of arbitrary characteristic.