Absolutely minimum attaining closed operators

TL;DR

This paper introduces and explores absolutely minimum attaining unbounded operators, establishing their properties, relationship with norm attaining operators, and characterizing injective cases with compact generalized inverses, revealing their hyperinvariant subspaces.

Contribution

It defines absolutely minimum attaining operators, characterizes injective cases via compact generalized inverses, and compares these with norm attaining bounded operators.

Findings

Injective absolutely minimum attaining operators have compact generalized inverses.

Every such operator possesses a non-trivial hyperinvariant subspace.

The paper establishes a relationship between unbounded and bounded norm attaining operators.

Abstract

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 (possibly unbounded) and characterize injective absolutely minimum attaining operators as those with compact generalized inverse. We give several consequences, one of them is that every such operator has a non trivial hyperinvariant subspace.