Abstract. Let $V$ be an infinite-dimensional vector space over a field. In a previous article [dSPSum4], we have shown that every endomorphism of $V$ splits into the sum of four square-zero ones but also into the sum of four idempotent ones. Here, we study decompositions into sums of three endomorphisms with prescribed split annihilating polynomials with degree $2$. Except for endomorphisms that are the sum of a scalar multiple of the identity and of a finite-rank endomorphism, we achieve a simple characterization of such sums. In particular, we give a simple characterization of the endomorphisms that split into the sum of three square-zero ones, and we prove that every endomorphism of $V$ is a linear combination of three idempotents.
DOI: 10.14232/actasm-016-319-1
AMS Subject Classification
(1991): 15A24, 16B50
Keyword(s):
infinite-dimensional vector space,
endomorphism,
decomposition,
square-zero endomorphism,
idempotent
Received November 18, 2016. (Registered under 69/2016.)
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