Solutions To The System Of Operator Equations AXB = C = BXA

The study on operator equations is one of the hot topics of functional analysis because of its multiple applications in different areas,for example,control theory,information theory and linear system theory.In the past decades,many scholars have devoted themselves to the research of different types of operator equations,and have made much progress.In this thesis,it is mainly considered the necessary and sufficient conditions for the existence of solutions to the system of operator equations AXB = C = BXA on a Hilbert space,and obtained the general forms of solutions to this system.Meanwhile,we make a further research into the operator equations AX = C,XB = D.We have the following contents:In the first part,the paper use“Douglas Range Inclusion Theorem" and Moore-Penrose inverse give the necessary and sufficient conditions for the existence of solu-tions,hermitian solutions and positive solutions to the operator equation AX = C with arbitrary operator A ? B(H).Moreover,we get the general forms of solutions,hermitian solutions and positive solutions to the operator equation AX = C.Af-ter that,we present some necessary and sufficient conditions for the existence of solutions to the system of operator equations AX = C,XB = D in the setting of A ? B(H)is arbitrary.Furthermore,we acquire the general forms of solutions to the system.In the second part,we obtain some necessary and sufficient conditions for the existence of solutions,hermitian solutions and positive solutions to the system of operator equations AXB = C=BXA by means of star partial order,and establish a relationship between the system AXB = C= BXA and the system BX = CAt,XB = AtC.Based on that,block-operator technique is used to characterize the general forms of hermitian solutions and positive solutions to this system.