  To study the compactness of bilinear commutators of certain bilinear
Calder\'on--Zygmund operators which include (inhomogeneous) Coifman--Meyer
bilinear Fourier multipliers and bilinear pseudodifferential operators as
special examples, Torres and Xue [Rev. Mat. Iberoam. 36 (2020), 939--956]
introduced a new subspace of BMO$\,(\mathbb{R}^n)$, denoted by
XMO$\,(\mathbb{R}^n)$, and conjectured that it is just the space
VMO$\,(\mathbb{R}^n)$ introduced by D. Sarason. In this article, the authors
give a negative answer to this conjecture by establishing an equivalent
characterization of XMO$\,(\mathbb{R}^n)$, which further clarifies that
XMO$\,(\mathbb{R}^n)$ is a proper subspace of VMO$\,(\mathbb{R}^n)$. This
equivalent characterization of XMO$\,(\mathbb{R}^n)$ is formally similar to the
corresponding one of CMO$\,(\mathbb{R}^n)$ obtained by A. Uchiyama, but its
proof needs some essential new techniques on dyadic cubes as well as some
exquisite geometrical observations. As an application, the authors also obtain
a weighted compactness result on such bilinear commutators, which optimizes the
corresponding result in the unweighted setting.
