Let T : D(T ) → H 2 be a densely defined closed operator with domain D(T ) ⊂ H 1 . We say T to be absolutely minimum attaining if for every closed subspace M of H 1 , the restriction operatorIn this article show that T is absolutely minimum attaining and unbounded if and only if its null space is finite dimensional and its Moore-Penrose inverse is compact. We also prove a Spectral Theorem for self -adjoint unbounded operators of this class. We show that every operator with a compact generalized inverse has a nontrivial hyperinvariant subspace.
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