Real Positive And Positive Solutions To The Operator Equation AXB=C

The purpose of this paper is, in the general setting of the adjointable operators between the Hilbert C^* -modules, to provide a new approach to the study of the real positive and positive solutions to the operator equation AXB = C. In the case that the underlying space is finite-dimensional, many results of the real positive solution[[8],[49]] and positive solution[[6],[59]]to the equation AXB = C have been given. Before we obtain general expression of the solution to the equation AXB = C, We study some equations, such as AXB^*- BX^*A^* = C, AXB = C B^*H^(+,*)(X) B≥0, and AXB = C B^*XB≥0. Then we apply the results and propose the necessary and sufficient conditions for the existence of a solution to the equation AXB = C, and obtain the general expression of the solution in the solvable case.The paper includes five chapters.In chapter 1, we will recall some knowledge about the Hilbert C^* -modules and Moore-Penrose inverse. In chapter 2, we we will study the general solutions to the operator equation AXB^* - BX^*A^* = C. In chapter 3, we will obtain the general expression of the real positive solution to the equation AXB = C. In chapter 4, we will give the necessary and sufficient conditions for the existence of a solution X to the operator equation AXB = C, B^*XB≥0, and provide a formula for the general solution to this operator equation. In chapter 5, we propose new necessary and sufficient conditions for the existence of a positive solution to the operator equation AXB = C, and derive new formula in each case for the general positive solution to this operator equation.