aspg-40-3285-1792469552 20261020041232 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 2024 4 2 Mohammed Noori Joudah Technical Engineering Department of Cooling and Air conditioning, Imam Jaafar A-Sadiq University, ThiQar, Iraq Emad Farhood Muhi Department of Accounting Techniques, ThiQar Technical College, Southern Technical University, ThiQar, Iraq 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 19 25 3285 https://creativecommons.org/licenses/by/4.0/ 10.54216/PMTCS.040203 https://www.americaspg.com/journal/40/article/3285 https://www.americaspg.com/storage/articles/manuscripts/01M1AFTY24S0RPPK8YSKA5MD4W.pdf