Monomial Rota-Baxter operators of weight zero and averaging operators on the polynomial algebra

Starting with the work S.H. Zheng, L. Guo and M. Rosenkranz (2015), Rota-Baxter operators are studied on the polynomial algebra. Injective Rota-Baxter operators of weight zero on $F[x]$ were described in 2021. We classify the following classes of monomial Rota-Baxter operators of weight zero on the polynomial algebra $F[x,y]$ and its augmentation ideal $F_0[x,y]$: 1) non-increasing in degree that do not contain monomials in the kernel, 2) mapping all monomials to themselves with a coefficient. In the context of these sets of operators, we show how one may define a monomial averaging operator by a given RB-operator and vice versa.