On the operator equation AX-XB=C
Salah Mecheri, Ahmed Bachie, A. Segress
TL;DR
This work investigates the generalized commutator problem in $B(H)$, focusing on the equation $AX - XB = C$ and its normal/Fuglede–Putnam settings. It establishes solvability criteria for $NX - XA = C$ and $AX - XN = C$ when one operator is normal and the pair has the $(FP)_{B(H)}$ property, via similarity of $2\times2$ block operators and explicit formulas, including a contour-integral representation under disjoint spectra. The paper further extends these ideas to equations like $AXB - CXD = E$ and proposes a Monkeypox-disease modeling framework $A^{*}XA + t AXA = Y$, yielding solutions through contour integrals when spectra are separable, with a special case $A=I$ giving $X = \frac{1}{t+1}Y$. Finally, it outlines open problems on commutator approximation in Schatten classes and on the geometry of the range of commutators, highlighting invariance properties under unitary equivalence and similarity and potential implications for quantum theory and applied 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.