Decompositions of linear transformations over division rings (Q2909074)

The author proves that for any endomorphism \(\gamma\) of a countable dimensional vector space \(V\) over a division ring \(D\) with more than \(3\) elements, there exist automorphisms \(\alpha\) and \(\beta\) of \(V\) such that \(\gamma-\alpha\), \(\gamma-\alpha^{-1}\), and \(\gamma^2-\beta^2\) are automorphisms of \(V\). In particular, this expresses \(\gamma\) as a sum of two automorphisms in two ways simultaneously. The result generalizes most of the countable dimensional case of a well known theorem of \textit{D. Zelinsky} [Proc. Am. Math. Soc. 5, 627--630 (1954; Zbl 0056.11002)], who proved that, aside from the case of the identity map on a 2-element vector space, every endomorphism of a vector space over a division ring is a sum of two automorphisms. The author also proves that for \(V\), \(D\), and \(\gamma\) as above, there exists an automorphism \(\alpha\) of \(V\) such that \(\gamma+\alpha\) and \(\gamma-\alpha^{-1}\) are automorphisms (cf. [\textit{H. Chen}, Glasg. Math. J. 52, No. 3, 427--433 (2010; Zbl 1218.15002)]).