Let T: D(T) → H2 be a densely defined closed operator with domain D(T) ⊂ H1. We say T to be absolutely minimum attaining if for every non-zero closed subspace M of H1 with D(T) ∩ M ≠ {0}, the restriction operator T|M: D(T) ∩ M → H2 attains its minimum modulus m(T|M). That is, there exists x ∈ D(T) ∩M with ‖x‖ = 1 and ‖T (x)‖ = inf{‖T (m)‖: m ∈ D(T) ∩ M: ‖m‖ = 1}. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. One such important characterization is that an unbounded operator belongs to this class if and only if its null space is finite dimensional and its Moore-Penrose inverse is compact. We also prove a spectral theorem for unbounded normal operators of this class. It turns out that every such operator has a compact resolvent. © D l, Zagreb.