A characterization of commutators of idempotents
Let \(R\) be a ring with the unit \(1\) in which the element \(2\) is invertible. The authors prove that an element \(t\in R\) is a commutator of a pair of idempotents if and only if there exist \(u\in R\) and \(s\in R\) such that \(u2=1\), \(ut+tu=0\), \(us=su\), \(st=ts\) and \(s2=t2+\frac{1}{4}\). This result is then used to prove that a necessary condition for a linear operator \(T\) (on a vector space) to be a commutator of a pair of idempotents is that it be similar to \(-T\), and that if \(T\) is a normal operator on a Hilbert space then this condition is both necessary and sufficient. In the special case of finite dimensional operators \(T\), \(T\) is a commutator of a pair of idempotents if and only if \(T\) is similar to \(-T\) and the Riesz part \(T_1\) of \(T\) corresponding to \(\frac{1}{2}i\) has the property that \(T2_1+\frac{1}{4}I\) has a square root commuting with \(T_1\).
scientific article; zbMATH DE number 1790226