SUMS OF SQUARE-ZERO OPERATORS

Abstract

This paper is concerned with characterizations of bounded linear operators on a complex Hilbert space which are expressible as a sum of two or more square-zero operators. We characterize sums of two square-zero operators among invertible operators, normal operators and operators on a finite-dimensional space. In particular, we show that if T is such a sum, then T and -T have the same spectra modulo the maximal ideal in the algebra of all bounded linear operators. This, together with a result of Pearcy and Topping's, yields a characterization of sums of four square-zero operators: T is such a sum if and only if it is a commutator. We also obtain various necessary or sufficient conditions for sums of three square-zero operators on a finite-dimensional space.