Absolutely minimum attaining closed operators

We define and discuss properties of the class of unbounded operators which attain minimum modulus. We establish a relationship between this class and the class of norm attaining bounded operators and compare the properties of both. Also we define absolutely minimum attaining operators (for not necessarily bounded) and characterize injective absolutely minimum attaining operators as those with compact generalized inverse. We give several consequences, one of those is that every such operator has a non trivial hyperinvariant subspace.