Conditions Implying Commutativity of Unbounded Self-adjoint Operators and Related Topics

TL;DR

This paper explores conditions under which unbounded self-adjoint operators commute, demonstrating that normality implies self-adjointness and establishing criteria for operator commutativity in quantum mechanics.

Contribution

It provides new characterizations of when unbounded self-adjoint operators commute based on normality conditions, using distinct approaches for each problem.

Findings

Normality of BA implies both BA and AB are self-adjoint.

Normality of AB implies BA is essentially self-adjoint.

Operator commutativity is characterized by normality conditions.

Abstract

Let $B$ be a bounded self-adjoint operator and let $A$ be a nonnegative self-adjoint unbounded operator. It is shown that if $BA$ is normal, it must be self-adjoint and so must be $AB$. Commutativity is necessary and sufficient for this result. If $AB$ is normal, it must be self-adjoint and $BA$ is essentially self-adjoint. Although the two problems seem to be alike, two different and quite interesting approaches are used to tackle each one of them.