  Let ${\mathcal X}$ be a metric space with doubling measure, $L$ a nonnegative
self-adjoint operator in $L^2({\mathcal X})$ satisfying the Davies-Gaffney
estimate, $\omega$ a concave function on $(0,\infty)$ of strictly lower type
$p_\omega\in (0, 1]$ and $\rho(t)={t^{-1}}/\omega^{-1}(t^{-1})$ for all $t\in
(0,\infty).$ The authors introduce the Orlicz-Hardy space
$H_{\omega,L}({\mathcal X})$ via the Lusin area function associated to the heat
semigroup, and the BMO-type space ${\mathop\mathrm{BMO}_{\rho,L}(\mathcal X)}$.
The authors then establish the duality between $H_{\omega,L}({\mathcal X})$ and
$\mathrm{BMO}_{\rho,L}({\mathcal X})$; as a corollary, the authors obtain the
$\rho$-Carleson measure characterization of the space
${\mathop\mathrm{BMO}_{\rho,L}(\mathcal X)}$. Characterizations of
$H_{\omega,L}({\mathcal X})$, including the atomic and molecular
characterizations and the Lusin area function characterization associated to
the Poisson semigroup, are also presented. Let ${\mathcal X}={\mathbb R}^n$ and
$ L=-\Delta+V$ be a Schr\"odinger operator, where $V\in
L^1_{\mathrm{\,loc\,}}({\mathbb R}^n)$ is a nonnegative potential. As
applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ is
bounded from $H_{\omega,L}({{\mathbb R}^n})$ to $L(\omega)$; moreover, if there
exist $q_1,\,q_2\in (0,\infty)$ such that $q_1<1<q_2$ and
{\normalsize$[\omega(t^{q_2})]^{q_1}$} is a convex function on $(0,\infty)$,
then several characterizations of the Orlicz-Hardy space
$H_{\omega,L}({{\mathbb R}^n})$, in terms of the Lusin-area functions, the
non-tangential maximal functions, the radial maximal functions, the atoms and
the molecules, are obtained. All these results are new even when
$\omega(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.
