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 $p\in(1, \infty)$.
Categories: math.CA