Complex Jordan operations
An operator J on a Hilbert space is called a complex Jordan operator if \(J=M+N\) where M is normal, \(N^ 2=0\) and \(MN=NM.\) If M is selfadjoint, then J is a Jordan operator. \textit{J. W. Helton} [Trans. Am. Math. Soc. 170, 305-331 (1972; Zbl 0251.47031)] proved that J is Jordan if and only if it satisfies the two equations \(J^{*^ 3}-3J^{*^ 2}J+3J^*J^ 2-J^ 3=0\) and \(J^ 3-3J^ 2J^*+3JJ^{*^ 2}-J^{*^ 3}=0.\) In the paper under review the author gives an algebraic characterization of complex Jordan operators: An operator J is complex Jordan if and only if it satisfies the three equations \[ J^{*^ n}J^3 - 3JJ^{*^ n}J^2 + 3J^2J^{*^ n}J - J^3 J^{*^ n} = 0, \quad n=1,2\text{ and }3. \] The result is applied to give another proof of Helton's characterization of Jordan operators. The author also discusses the independence of the three equations in his characterization of complex Jordan operators.
scientific article; zbMATH DE number 3844443