## 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.)

### Citations:

Zbl 0887.15024; Zbl 1081.15018
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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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