Let
$(X,d,\unicode[STIX]{x1D707})$
be a metric measure space endowed with a distance
$d$
and a nonnegative, Borel, doubling measure
$\unicode[STIX]{x1D707}$
. Let
$L$
be a nonnegative self-adjoint operator on
$L^{2}(X)$
. Assume that the (heat) kernel associated to the semigroup
$e^{-tL}$
satisfies a Gaussian upper bound. In this paper, we prove that for any
$p\in (0,\infty )$
and
$w\in A_{\infty }$
, the weighted Hardy space
$H_{L,S,w}^{p}(X)$
associated with
$L$
in terms of the Lusin (area) function and the weighted Hardy space
$H_{L,G,w}^{p}(X)$
associated with
$L$
in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duong et al. [‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’, J. Geom. Anal.
26 (2016), 1617–1646], who proved that
$H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$
for
$p\in (0,1]$
and
$w\in A_{\infty }$
by imposing an extra assumption of a Moser-type boundedness condition on
$L$
. Our result is new even in the unweighted setting, that is, when
$w\equiv 1$
.