Singular Lyapunov operator equations: applications to C∗- algebras, Fréchet derivatives and abstract Cauchy problems

Abstract

Let A be a closed operator on a separable Hilbert space 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.