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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.

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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