Bauer, B., & Gerhold, S. (2024). A characterization of real matrix semigroups. Research in Mathematics, 11(1), Article 2289203. https://doi.org/10.1080/27684830.2023.2289203
We characterize all real matrix semigroups, indexed by the non-negative reals, which satisfy a mild boundedness assumption, without assuming continuity. Besides the continuous solutions of the semigroup functional equation, we give a description of solutions arising from non-measurable solutions of Cauchy’s functional equation. To do so, we discuss the primary decomposition and the Jordan—Chevalley decomposition of a matrix semigroup. Our motivation stems from a characterization of all multi-dimensional self-similar Gaussian Markov processes, which is given in a companion paper.
