Spectral representation of absolutely minimum attaining unbounded normal operators

Let $T:D(T)\to H_2$ 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 $T|M:D(T)\cap M\rightarrow H_2$attainsitsminimummodulus$m(T|{M})$. That is, there exists $x\in D(T)\cap M$ with $\mathrm{\parallel }x\mathrm{\parallel }=1$ 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.