Let
λ > 0 and
$${Delta _lambda }: = - frac{{{d^2}}}{{d{x^2}}} - frac{{2lambda }}{x}frac{d}{{dx}}$$
be the Bessel operator on R
+:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R
+,
x2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function
b ∈ BMO(R
+,
x2λdx) is in CMO(R
+,
x2λdx) if and only if the Riesz transform commutator xxxx is compact on
Lp(R
+,
x2λdx) for all
p ∈ (1,∞).