Linear operators that preserve pairs of matrices which satisfy extreme rank properties -- a supplementary version.
A pair of \(m\times n\) matrices \((A,B)\) is said to be rank-sum-maximal if \(p(A+ B)= p(A)+ p(B)\), and rank-sum-minimal if \(p(A+ B)=| p(A)- p(B)|\). The author characterizes the linear operators preserving the set of rank-sum-maximal pairs over any field and the linear operators preserving the set of rank-sum-maximal pairs are characterized by using a result about rank preservers proposed by \textit{C.-K. Li} and \textit{S. Pierce} [Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)], and thereby the linear preservers of the set of rank-sum-minimal pairs are characterized. The paper can be viewed as a supplementary version of several related results. Suppose \(F\) is an arbitrary field. Let \(M_{m,n}(F)\) be the set of all \(m\times n\) matrices over \(F\). Let \(\theta_+\) (respectively, \(\theta_-\)) be the subset of \(M_{m,n}(F)\times M_{m,n}(F)\) consisting of all rank-sum-maximal (respectively, rank-sum-minimal) pairs. Section 2 characterizes the linear preservers of the set \(\theta_+\) over any field \(F\). Section 3 characterizes the linear preservers of the set \(\theta_-\) over an field \(F\) except for \(\{0\}\).
scientific article; zbMATH DE number 2013059