Let A be a closed operator on a separable Hilbert space \(\mathcal {H}\). In this paper we obtain sufficient conditions for the existence of a solution to the Lyapunov operator equation \( A^*X+X^*A=I\), under the assumption that it is singular (without a unique solution). Specially, if A is a self-adjoint operator, we derive sufficient conditions for the solution X to be symmetric. We also show that these results hold in the bounded-operator setting and in \(C^*-\)algebras. By doing so, we generalize some known results regarding solvability conditions for algebraic equations in \(C^*-\)algebras. We apply our results to study some functional problems in abstract analysis.

设A是可分希尔伯特空间\(\mathcal {H}\)上的闭算子。在本文中，我们获得了李雅普诺夫算子方程\( A^*X+X^*A=I\)存在解的充分条件，假设它是奇异的（没有唯一解）。特别地，如果A是自伴随算子，我们推导出解X对称的充分条件。我们还表明，这些结果适用于有界算子设置和\(C^*-\)代数。通过这样做，我们概括了一些关于\(C^*-\) 中代数方程的可解性条件的已知结果代数。我们应用我们的结果来研究抽象分析中的一些功能问题。