Strong Commutativity of Unbounded Self-adjoint Operators on a Separable Hilbert space

Fidelis Mukudi ,Lucy Chikamai ,J K Mile ,Shem Aywa

doi:10.7176/mtm/10-8-03

Abstract

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

Highlights

The unbounded Self-adjoint operators are said to be commute strongly if their bounded transforms commute
It is known that strongly commuting unbonded Self-adjoint operators that commute on a common core commute pointwise, the pointwise commuting unbounded Self-adjoint operators do not necessarily commute strongly
This was discovered by Nelson in [4] when he sited an example of a pair unbounded Selfadjoint operators that commute pointwise on a common core but do not commute strongly

Summary

Introduction

The unbounded Self-adjoint operators are said to be commute strongly if their bounded transforms commute. We show that by making the underlying space be a separable Hilbert Space and suitably selecting a special positive operator, the pointwise commutating unbounded Self-adjoint operators commute strongly. The result states in part that if A is a closed and densely defined Dirichlet operator on a square-integrable space that is associated with a uniformly elliptic differential operator, there exists a sufficiently large α such that A α exists given that α α ∈ R. This is a case of unbounded operators with compact inverses. Using the functional calculus for self-adjoint operators, the spectral representation for A α and B α are

The functions f λ

FEx I P y

Aα λ α dP λ