  Let $T:D(T)\rightarrow H_2$ be a densely defined closed operator with domain
$D(T)\subset H_1$. We say $T$ to be absolutely minimum attaining if for every
closed subspace $M$ of $H_1$, the restriction operator $T|_M:D(T)\cap
M\rightarrow H_2$ attains its minimum modulus $m(T|_{M})$. That is, there
exists $x \in D(T)\cap M$ with $\|x\|= 1$ and $\|T(x)\| = \inf \{\|T(m)\|: m
\in D(T) \cap M: \|m\|=1\}$. 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. We also prove a spectral
theorem for unbounded normal operators of this class. It turns out that every
such operator has a compact resolvent.
