Some Investigations On The Solutions To Some Systems Of Operator Equations

It is a very active research topic to investigate matrix equations or oper-ator equations by generalized inverse in the past decade. In fact, it has been extensively studied by a number of authors. In this dissertation, we study the solutions to some system of operator equations or system of matrix equations. We get many meaningful new results which extend some known results. These results further enrich and develop the matrix algebra and the operator algebra.The dissertation is divided into 5 chapters.In Chapter 1 the research background as well as the main contributions of this dissertation are introduced. The background of this paper and the basic material for later use are also presented.In Chapter 2 we investigate the general solutions to operator equations AX*+XA*= B, AX-X*C= B and the general solution with special structure to operator equation.AXA*= B in the Hilbert space.In Chapter 3 we establish necessary and sufficient conditions for the existence and the explicit expression of the general solution to the system of adjointable operator equations A1X= D1,XB2= D2,A3XB3+B3*X*C3=D3 over the Hilbert C*-modules. As an application, we discuss the anti-reflexive Hermitian solution to the system of complex matrix equations AX=B, XC=D, EXE*= F.In Chapter 4 we consider the positive solution to the system of adjointable operator equations A1X=C1; XB2=C2, A3XB3=C3 over Hilbert C*-modules. Using the results, we give the positive solution to complex matrix equation AXB=C.In Chapter 5 we investigate the positive solution to the system of equations A1X1=C1,X1B1=D1,A2X2=C2,X2B2=D2,A3X1A3*+A4X2A4*=C5 for adjointable operators between Hilbert C*-modules. As an application, we consider the reflexive positive semidefinite solution to the system of complex mattix equations.AX=B,XC=D,EXE*=F,...