Linear preservers for Sylvester and Frobenius bounds on matrix rank

Let A and B be n × n matrices. A classical result about the rank function is Sylvester's inequality which states that the rank of the product of AB is at most min{rank (A), rank (B)} and at least rank (A) + rank (B) - n. A generalization of Sylvester's inequality is Frobenius's inequality which states that rank (AB) + rank (BC) ≤ rank (ABC) + rank (B). In this paper we investigate the structure of linear operators that preserve those ordered pairs or triples of matrices which satisfy one of the extreme cases in these inequalities.