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 | 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. |
Keywords: | Abstract Cauchy problems | Equations in C - algebras ∗ | Fréchet derivative | Lyapunov operator equations | Publisher: | Springer Link |
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