We investigate the Hardy space
H
L
1
H^1_L
associated with a self-adjoint operator
L
L
defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an
L
L
-harmonic non-negative function
h
h
such that the semigroup
exp
(
−
t
L
)
\exp (-tL)
, after applying the Doob transform related to
h
h
, satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space
H
L
1
H^1_L
in terms of a simple atomic decomposition associated with the
L
L
-harmonic function
h
h
. Our approach also yields a natural characterisation of the
B
M
O
BMO
-type space corresponding to the operator
L
L
and dual to
H
L
1
H^1_L
in the same circumstances.
The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in
R
n
{\mathbb {R}^n}
, Schrödinger operators with certain potentials, and Bessel operators.