STRONG COMMUTATIVITY OF UNBOUNDED SELF-ADJOINT OPERATORS ON A SEPARABLE HILBERT SPACE

Abstract

Commutativity is an important concept in mathematical physics, nuclear energy generation and related fields. In the study and exploration of particles in atoms and molecules, the observables of these a particle that is, the position, the momentum and the spin are expressed as special functions called unbounded Self-adjoint operators. When these operators commute with another, then it becomes easy to compute the measurements of one operator given that of another because they share the same eigenstate. The measurements may be the eigenvalues and the spectral measures among others. The commutativity of operators, may be regarded as either pointwise or strong. When operators commute strongly, their spectral measure and bounded transforms commute as well. 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 no guarantee that they will commute strongly. By imposing some conditions, 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.