Spectral representation of absolutely minimum attaining unbounded normal operators

TL;DR

This paper characterizes absolutely minimum attaining unbounded normal operators, proves they have nontrivial hyperinvariant subspaces, and establishes a spectral theorem showing such operators have a compact resolvent.

Contribution

It provides a spectral theorem for unbounded normal operators that are absolutely minimum attaining, revealing their spectral properties and hyperinvariant subspace structure.

Findings

Operators in this class have a nontrivial hyperinvariant subspace.

Every such operator has a compact resolvent.

A spectral theorem is established for these unbounded normal operators.

Abstract

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.