LAUSR

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.