  We consider the abstract non-negative self-adjoint operator $L$ acting on
$L^2(X)$ which satisfies Davies-Gaffney estimates and the corresponding Hardy
spaces $H^p_L(X)$. We assume that doubling condition holds for the metric
measure space $X$. We show that a sharp H\"ormander-type spectral multiplier
theorem on $H^p_L(X)$ follows from restriction type estimates and the
Davies-Gaffney estimates. We also describe the sharp result for the boundedness
of Bochner-Riesz means on $H^p_L(X)$.
