  Let $(\mathcal{X}, d, \mu)$ be a metric measure space, $L$ a linear operator
which has a bounded $H_\infty$ functional calculus and satisfies the
Davies-Gaffney estimate, $\Phi$ a concave function on $(0,\infty)$ of critical
lower type $p_\Phi^-\in(0,1]$ and $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for
all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO
space ${\mathrm {VMO}}_{\rho,L}({\mathcal X})$ associated with $L$, and
establish its characterization via the tent space. As applications, the authors
show that $({\mathrm {VMO}}_{\rho,L}({\mathcal X}))^*=B_{\Phi,L^*}({\mathcal
X})$, where $L^*$ denotes the adjoint operator of $L$ in $L^2({\mathcal X})$
and $B_{\Phi,L^*}({\mathcal X})$ the Banach completion of the Orlicz-Hardy
space $H_{\Phi,L^*}({\mathcal X})$.
