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The spectral characterization of generalized projections. (English) Zbl 1067.47001

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 J. Groß and 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 J. K. Baksalary and 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.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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References:

[1] Baksalary, J.K.; Baksalary, O.M., On linear combinations of generalized projectors, Linear algebra appl., 388, 17-24, (2004) · Zbl 1081.15016
[2] Baksalary, J.K.; Liu, X., An alternative characterization of generalized projectors, Linear algebra appl., 388, 61-65, (2004) · Zbl 1081.15018
[3] Conway, J.B., A course in functional analysis, (1990), Springer-Verlag · Zbl 0706.46003
[4] Groß, J.; Trenkler, G., Generalized and hypergeneralized projectors, Linear algebra appl., 364, 463-474, (1997) · Zbl 0887.15024
[5] Kadison, R.V.; Ringrose, J.R., Fundamentals of the theory of operator algebras, vol. I, (1983), Academic Press · Zbl 0518.46046
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