A localization technique for linear preserver problems

Leiba Rodman, William & Mary
Peter Semrl

Abstract

We introduce a new general technique for solving linear preserver problems. The idea is to localize a given linear preserver phi at each nonzero vector. In such a way we get vector-valued linear maps on the space of matrices which inherit certain properties from phi. If we can prove that such induced maps have a standard form, then phi itself has either a standard form or a very special degenerate form. We apply this technique to characterize linear preservers of full rank. Using this technique we further reprove two classical results describing the general form of linear preservers of rank one and linear preservers of the unitary (or orthogonal) group. (C) 2009 Published by Elsevier Inc.