Authors: Đorđević, Bogdan 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Singular Lyapunov operator equations: applications to C∗- algebras, Fréchet derivatives and abstract Cauchy problems
Journal: Analysis and Mathematical Physics
Volume: 11
Issue: 4
First page: 160
Issue Date: 1-Dec-2021
Rank: ~M21a
ISSN: 1664-2368
DOI: 10.1007/s13324-021-00596-z
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.
Keywords: Abstract Cauchy problems | Equations in C - algebras ∗ | Fréchet derivative | Lyapunov operator equations
Publisher: Springer Link

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