  Let $(X, d, \mu)$ be a space of homogeneous type, i.e. the measure $\mu$
satisfies doubling (volume) property with respect to the balls defined by the
metric $d$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume
that the semigroup of $L$ satisfies the Davies-Gaffney estimates. In this
paper, we study the weighted Hardy spaces $H^p_{L,w}(X)$, $0 < p \le 1$,
associated to the operator $L$ on the space $X$. We establish the atomic and
the molecular characterizations of elements in $H^p_{L,w}(X)$. As applications,
we obtain the boundedness on $\HL$ for the generalized Riesz transforms
associated to $L$ and for the spectral multipliers of $L$.
