General solutions to equation $axb^*-bx^*a^*=c$ in rings with involution

TL;DR

This paper generalizes the solutions to a specific operator equation involving ring elements with involution, extending previous results from Hilbert C*-modules and matrices to a broader ring setting.

Contribution

It provides a unified solution framework for the equation in rings with involution, broadening the scope beyond operators and matrices.

Findings

Derived necessary and sufficient conditions for solutions in rings with involution.

Unified approach encompassing matrices and operators as special cases.

Extended previous operator equation solutions to a more general algebraic setting.

Abstract

In [Q. Xu et al., The solutions to some operator equations, Linear Algebra Appl.(2008), doi:10.1016/j.laa.2008.05.034], Xu et al. provided the necessary and sufficient conditions for the existence of a solution to the equation $AXB^*-BX^*A^*=C$ in the general setting of the adjointable operators between Hilbert $C^*$-modules. Based on the generalized inverses, they also obtained the general expression of the solution in the solvable case. In this paper, we generalize their work in the more general setting of ring $R$ with involution * and reobtain results for rectangular matrices and operators between Hilbert $C^*$-modules by embedding the "rectangles" into rings of square matrices or rings of operators acting on the same space.