In this paper, we study and discuss the existence and uniqueness of the solution of the discretetime 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,, and tA (x)= A * X+tXA, XB(H).for A,B -1 and A -1 are fixed operators in B(H),and t is any scalar.
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Kuffi et al. (2007) studied this question.