Linear operators that preserve pairs of matrices which satisfy extreme rank properties (Q1611887)

Let \({\mathbb F}\) be a field with at least \(m+2\) elements and of characteristic not~\(2\). Then a linear operator \(T:M_{m,n}({\mathbb F})\to M_{m,n}({\mathbb F})\) preserves the set of all rank-sum-maximal (rank-sum-minimal) pairs of matrices if and only if either \(T\equiv 0\) or \(T\)~is a \((U,V)\)-operator. This theorem generalizes results by \textit{L. B. Beasley} [Linear operators which preserve pairs on which the result is additive. J. Korean SIAM, 2, 27-30 (1998)] and \textit{A.~Guterman} [Linear Algebra Appl. 331, 75-87 (2001; Zbl 0985.15018)]. Here a pair \(A,B\in M_{m,n}({\mathbb F})\) is called rank-sum-maximal (rank-sum-minimal) if \(\text{rank}(A+B)= \text{rank}(A)+\text{rank}(B)\) (\(\text{rank} (A+B)=|\text{rank}(A)-\text{rank}(B)|)\). A~linear operator \(T\) is a \((U,V)\)-operator if there exist invertible matrices \(U,V\) such that \(T(A)=UAV\) for all \(A\in M_{m,n}({\mathbb F})\) or, if \(m=n\), \(T(A)=UA^tV\) for all \(A\in M_{m}({\mathbb F})\).