Some general techniques on linear preserver problems (Q1583717)

Linear preserver problems are those of the following kind: characterize the linear mappings of one vector space of matrices into another that preserve given matrix properties (such as the rank or being symmetric). After listing various general techniques that have been devised for attacking such problems, the authors put forward three more of their own. The first relies on the ``transfer'' principle from model theoretic algebra which states that if a first-order property holds in one algebraically closed field of given characteristic then it holds in them all. A typical application is to show that the characterization of rank-preserving linear mappings obtained over the complex field is valid over an arbitrary algebraically closed field of characteristic 0. The second technique relies on a result of the following kind: if a linear mapping \(\varphi\) satisfies \(\varphi(S)\subseteq S\) for a given set \(S\) of matrices, then it also satisfies \(\varphi(T_r) \subseteq T_r (r=0,1, \dots)\) for certain sets \(T_r\) geometrically related to \(S\). The idea is that information about the \(\varphi\) preserving one or more \(T_r\) yields information about the \(\varphi\) preserving \(S\). Various applications are given, some new. The third technique aims, in a similar way, to obtain information about a given problem from certain auxiliary problems in which idempotents are preserved. There is an interesting application to certain mappings of Banach spaces. Throughout the paper, the general methods are illustrated by a wealth of examples.