Matrix groups with positive spectra
A group \({\mathcal G}\) of operators acting on a real or complex vector space \({\mathcal V}\) is called divisible if for every \(A\in{\mathcal G}\) there exists an operator \(B\in{\mathcal G}\) such that \(B^ 2=A\). Now let \({\mathcal G}\) be a divisible group of operators with positive spectra acting on the vector space \({\mathcal V}\) such that \({\mathcal G}\) is relatively closed in the group \({\mathcal L}({\mathcal V})\) of all invertible operators on \({\mathcal V}\). Then there is a bijection \(\Phi:{\mathcal G}\to{\mathcal L}=\log({\mathcal G})\), and it is shown that \({\mathcal L}\) is a real exponentially invariant Lie algebra (i.e. \(e^ ABe^{-A}\in{\mathcal L}\) for all \(A,B\in{\mathcal L})\) whose elements have only real spectra. Since the commutators of \({\mathcal L}\) are nilpotent the Lie algebra \({\mathcal L}\) turns out to be triangularizable. Thus also \({\mathcal G}\) itself is triangularizable. Furthermore, the author gives a construction of irreducible matrix groups whose elements have positive trace. Examples show that the divisibility condition on \({\mathcal G}\) is necessary, and that the results on the triangularizability do not apply to semigroups of operators.
divisible group of operators with positive spectra
scientific article; zbMATH DE number 64137