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We say that an L^1-function f belongs to the Hardy space H^1_L if the maximal function M_L f(x)=\\sup_{t>0} |e^{-tL} f(x)| belongs to L^1(R^d). We prove that under certain assumptions on V the space H^1_L is also characterized by the Riesz transforms R_j=\\frac{\\partial}{\\partial x_j} L^{-1/2}, j=1,...,d, associated with L. 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