Let T : H 1 Ñ H 2 be a bounded linear operator defined between complex Hilbert spaces H 1 and H 2 . We say T to be minimum attaining if there exists a unit vector x P H 1 such that }T x} " mpT q, where mpT q :" inf t}T x} : x P H 1 , }x} " 1u is the minimum modulus of T . We say T to be absolutely minimum attaining (AM-operators in short), if for any closed subspace M of H 1 the restriction operator T | M : M Ñ H 2 is minimum attaining. In this paper, we give a new characterization of positive absolutely minimum attaining operators (AM-operators, in short), in terms of its essential spectrum. Using this we obtain a sufficient condition under which the adjoint of an AM-operator is AM. We show that a paranormal absolutely minimum attaining operator is hyponormal. Finally, we establish a spectral decomposition of normal absolutely minimum attaining operators. In proving all these results we prove several spectral results for paranormal operators. We illustrate our main result with an example.