Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 10.54216/PMTCS https://www.americaspg.com/journals/show/3285 2023 2023 Technical Engineering Department of Cooling and Air conditioning, Imam Jaafar A-Sadiq University, ThiQar, Iraq admin admin Department of Accounting Techniques, ThiQar Technical College, Southern Technical University, ThiQar, Iraq Emad Farhood Muhi This paper provides a method to solve the discrete time Lyapunov equation. Identified and discussed. If the equation takes the following form: D (λy+μz) = λDy+ μDz , 𝑦,z∈ y; λ ,μ ∈𝐹 . If ∃ a constant e∈∞ ∋ ||Dy|| ≤ e ||y||, y ∀Y. and D is bounded, then D is called a linear operator equation. In particular, (Lyapunov and Sylvester operator equations) are very important in differential equations, integral equations and many other branches of mathematics. The study of solutions and of the above equestion We also discussed operator equations and special kinds of operators and studied some elementary operators. These operators are generalizations of operators τ𝐴𝐷:𝐷(𝐻)→𝐷(𝐻) τ𝐴𝐷:𝜏𝐴𝐷(𝑦)=𝐴𝑦−𝑦𝐷, 𝑦∈𝐷(𝐻) 2024 2024 32 42 10.54216/PMTCS.040203 https://www.americaspg.com/articleinfo/40/show/3285