Solvability of The Operator Equations $ AX-XB=C $ and $ AX-YB=C $

TL;DR

This paper establishes new necessary and sufficient conditions for solving certain operator equations involving group invertible operators on infinite-dimensional Hilbert spaces, and derives their general solutions.

Contribution

It provides novel solvability criteria and explicit solutions for operator equations involving group invertible operators, extending previous results.

Findings

New solvability conditions for $AX-XB=C$ and $AX-YB=C$

Explicit general solutions in terms of group inverses

Extended criteria for $AYB-Y=C$ equation

Abstract

This paper provides new necessary and sufficient conditions for the solvability to the operator equations $ AX-XB=C$ and $AX-YB=C,$ where $A $ and $B $ are group invertible operators defined on an infinite dimensional Hilbert space. In addition, the general solutions to the equation $AX-YB=C,$ are derived in terms of group inverse of $ A $ and $ B $. As a consequence, new necessary and sufficient conditions for the solvability to the operator equation $ AYB-Y=C,$ are derived.