Compactness of Riesz transform commutator associated with Bessel operators

Let $\lambda>0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. 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 $b\in {\rm BMO}(\mathbb R_+,\, x^{2\lambda}dx)$ is in ${\rm CMO}(\mathbb R_+,\, x^{2\lambda}dx)$ if and only if the Riesz transform commutator $[b, R_{\Delta_\lambda}]$ is compact on $L^p(\mathbb R_+, x^{2\lambda}dx)$ for any $p\in(1, \infty)$.