Compactness of Riesz transform commutator associated with Bessel operators

Let $\lambda >0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$betheBesseloperatoron$\mathbb R_+:=(0,\infty)$. We first introduce and obtain an equivalent characterization of ${\rm CMO}(\mathbb R_+,, x^{2\lambda}dx)$. By this equivalent characterization and establishing a new version of the Fr'{e}chet-Kolmogorov theorem in the Bessel setting, we further prove that a function $b\in \mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}_{+},x^{2\lambda }dx)$ is in $\mathrm{C}\mathrm{M}\mathrm{O}({\mathbb{R}}_{+},x^{2\lambda }dx)$ if and only if the Riesz transform commutator $[b,R_{{\mathrm{\Delta }}_{\lambda }}]$ is compact on $L^p(\mathbb R_+, x^{2\lambda}dx)$forany$p\in(1, \infty)$.