Linear rank preservers on infinite triangular matrices (Q2794873)

Let \(\mathcal S\) be a space of matrices over a field. A mapping \(\phi:{\mathcal S}\to{\mathcal S}\) is a rank-\(k\) preserver if, for all \(x\in{\mathcal S}\), \(\mathrm{rank}\,(x)=k\) implies that \(\mathrm{rank}\,(\phi(x))=k\). Let \({\mathcal T}_\infty(F)\) denote the space of infinite upper triangular matrices over a field \(F\). (Rows and columns have indices \(1,2,3,\dots\).) The author proves that if a mapping \(\phi: {\mathcal T}_\infty(F)\to{\mathcal T}_\infty(F)\) is a linear rank-one preserver, then either (1)~\(\phi({\mathcal T}_\infty(F))\) consists only of rank-one matrices and the zero matrix or (2)~there exist matrices \(a\) and \(b\) such that \(\phi(x)=axb\) for all \(x\in{\mathcal T}_\infty(F)\). She also proves that an injection \(\phi:{\mathcal T}_\infty(F)\to{\mathcal T}_\infty(F)\) is a linear rank-\(k\) preserver, \(k\geq 2\), if and only if it is a linear rank-one preserver and \(\phi({\mathcal T}_\infty(F))\) is not a rank-one subspace of~\({\mathcal T}_\infty(F)\). Related results in the finite-dimensional case are well-known.