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Sums of square-zero operators. (English) Zbl 0745.47006
An operator $$T$$ is square-zero if $$T^ 2=0$$. The aim of this paper is to characterize bounded linear operators on complex Hilbert space which are expressible as a sum of two or more square-zero operators. The authors completely characterize such operators among invertible operators, normal operators and operators on finite-dimensional spaces. For noninvertible operators on an infinite-dimensional space they give various necessary and/or sufficient conditions for characterization of such operators. In particular, they show that if $$T$$ is the sum of two square-zero operators then $$T$$ and $$-T$$ have the same spectra. This result is the main ingredient in proving the above mentioned result of sums of four square- zero operators, e.g. the result: $$T$$ is such a sum if and only if it is a commutator.

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47B47 Commutators, derivations, elementary operators, etc. 47A65 Structure theory of linear operators
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