Operators, Spaces, Algebras, Modules 2010

Isomorphisms of quasi-commutativity relation

We say that the operators $A$ and $B$ quasi-commute if $AB$ and $BA$ are linearly dependent and either both zero or both nonzero. This includes the relation of commutativity and anti-commutativity. Recently, linear bijections which strongly preserve quasi-commutativity on complex matrices, or on its subspace of hermitian matrices, where classified by Molnar. His result was then extended by \v Semrl and Radjavi to bounded operators acting on a Banach space, assuming that quasi-commutativity is preserved in one direction only. Following the recent line of investigation in preserver problems, we were able to drop the assumption of linearity and classify bijective, possibly nonlinear maps which preserve quasi-commutativity relation in both directions.