The structure of linear preservers of gs-majorization (Q2460753)

A real or complex matrix is generalized doubly stochastic if all its row and column sums equal to 1. Contrary to the ordinary stochastic matrices, the entries are not assumed to be nonnegative. Motivated by a classic result -- see Theorem 46 in the book by \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya} [Inequalities. Cambridge UP (1934; Zbl 0010.10703)] -- about vector majorization the authors say a rectangular \(n\)-by-\(m\) matrix \(B\) is \(gs\)-majorized by \(A\), (denoted by \(A\succ B\)), if \(B=DA\) for some generalized doubly stochastic matrix \(D\). They proceed to classify linear maps \(T\) which preserve \(gs\)-majorization on rectangular real or complex matrices (that is, which satisfy \(A\succ B\Rightarrow T(A)\succ T(B)\)). The case of \(n\)-by-\(1\) rectangular matrices is also included. When a linear \(T\) preserves \(gs\)-majorization in both directions, a further reduction on the possible cases is obtained.