  Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a
one-to-one operator of type $\omega$ having a bounded $H_\infty$-functional
calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$
off-diagonal estimates on balls, where $p_L\in[1,2)$ and $q_L\in(2,\infty]$.
Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such
that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb
A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its
uniformly critical upper type index $I(\varphi)\in(0,1]$ and $\varphi(\cdot,t)$
satisfies the uniformly reverse H\"older inequality of order
$(q_L/I(\varphi))'$. In this paper, the authors introduce a
Musielak-Orlicz-Hardy space $H_{\varphi,\,L}(\mathcal{X})$, via the Lusin-area
function associated with $L$, and establish its molecular characterization. In
particular, when $L$ is nonnegative self-adjoint and satisfies the
Davies-Gaffney estimates, the atomic characterization of
$H_{\varphi,\,L}(\mathcal{X})$ is also obtained. Furthermore, a sufficient
condition for the equivalence between $H_{\varphi,\,L}(\mathbb{R}^n)$ and the
classical Musielak-Orlicz-Hardy space $H_{\varphi}(\mathbb{R}^n)$ is given.
Moreover, for the Musielak-Orlicz-Hardy space $H_{\varphi,\,L}(\mathbb{R}^n)$
associated with the second order elliptic operator in divergence form on $\rn$
or the Schr\"odinger operator $L:=-\Delta+V$ with $0\le V\in
L^1_{\mathrm{loc}}(\mathbb{R}^n)$, the authors further obtain its several
equivalent characterizations in terms of various non-tangential and radial
maximal functions; finally, the authors discuss the boundedness of the Riesz
transform $\nabla L^{-1/2}$.
