SOLUTION OF OPERATOR EQUATIONS

In this paper, we study and discuss the existence and uniqueness of the solution of the discrete – time Sylvester and Lyapunov operator equations, and more general continuous –time Lyapunov operator equations. Also, we study the nature of the solution for these operator equation for special types of operators, as well as the study of the range A,B, A,and tA are introduced. Where, A,B(x)=AX-XB-1 A(x)=A*X-XA-1, and tA(x)= A*X+tXA, XB(H). for A,B-1and A-1 are fixed operators in B(H),and t is any scalar.