  Let $L_{1}$ and $L_{2}$ be non-negative self-adjoint operators acting on
$L^{2}(X_{1})$ and $L^{2}(X_{2})$, respectively, where $X_{1}$ and $X_{2}$ are
spaces of homogeneous type. Assume that $L_{1}$ and $L_{2}$ have Gaussian heat
kernel bounds. This paper aims to study some equivalent characterizations of
the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$
associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$
belongs to the product Muckenhoupt class $A_{\infty}(X_{1} \times X_{2})$. Our
main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$
introduced via area functions can be equivalently characterized by
Littlewood-Paley $g$-functions, Littlewood-Paley $g^{\ast}_{\lambda_{1},
\lambda_{2}}$-functions, and Peetre type maximal functions, without any further
assumptions beyond the Gaussian upper bounds on the heat kernels of $L_{1}$ and
$L_{2}$. Our results are new even in the unweighted product setting.
