On the operator equation AX-XB=C

TL;DR

This paper explores the operator equation AX - XB = C, motivated by quantum theory problems, and demonstrates solutions with applications to related equations and an analogy to Monkeypox disease modeling.

Contribution

It provides a detailed analysis of solving the operator equation AX - XB = C and applies these solutions to more complex equations and disease modeling scenarios.

Findings

Solutions to the operator equation AX - XB = C are characterized.

Application of solutions to the equation AXB - CXD = E.

Illustration of mathematical methods applied to Monkeypox disease modeling.

Abstract

This work studies how certain problems in quantum theory have motivated some recent research in pure Mathematics in matrix and operator theory. The mathematical key is that of a commutator or a generalized commutator, that is, find an operator $X\in B(H)$ satisfying the operator equation $AX - XB = C$. By this we will show how and why to solve the operator equation $AX - XB = C$. As application we recall the solurion of $AXB-CXD=E$ and we use the solution of this equation to solve equation of Monkeypox Diseases.