Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results

TL;DR

This paper investigates properties of unbounded operators, focusing on their square roots, nilpotence, and invertibility, providing examples, counterexamples, and new insights into their structural characteristics.

Contribution

It introduces new examples and counterexamples of unbounded operators with specific properties, expanding understanding of their roots, closability, and invertibility.

Findings

Explicit examples of unbounded non-closable nth roots of the identity and zero operators

Existence of non-closable unbounded operators without non-closable square roots

Methods to find bijective, surjective without injective, and injective without surjective operators

Abstract

In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit everywhere defined unbounded non-closable $nth$ roots of the identity operator as well as the zero operator. We also show a non-closable unbounded operator without any non-closable square root. Among other consequences, we have a way of finding everywhere defined bijective operators, everywhere defined operators which are surjective without being injective and everywhere defined operators which are injective without being surjective. Some related results on nilpotence are also given.