The spectral characterization of generalized projections
A bounded operator \(T\) on a Hilbert space \(H\) is called a generalized projection if \(T^2=T^*\). This notion was first introduced in the finite-dimensional case by \textit{J. Groß} and \textit{G. Trenkler} [Linear Algebra Appl. 264, 463--474 (1997; Zbl 0887.15024)]. The authors of the paper under review prove that an operator \(T\) is a generalized projection if and only if sp\((T)\subseteq\{0,1,e^{\pm i\frac{2}{3}\pi}\}\). They also extend the main result of \textit{J. K. Baksalary} and \textit{X. Liu} [Linear Algebra Appl. 388, 61--65 (2004; Zbl 1081.15018)] to the infinite-dimensional case. In fact, they introduce a spectral representation for a generalized projection and then show that the following statements are equivalent: (i) \(T\) is a generalized projection; (ii) \(T^4=T\) and \(T\) is normal; (iii) \(T^4=T\) and \(T\) is a partial isometry.
scientific article; zbMATH DE number 2173032