Linear mappings on second symmetric product spaces that Preserve rank less than or equal to one

In this paper we characterize those linear mappings from a second symmetric product space to another which preserve decomposable elements of the form λu⋅u where u is a vector and λ is a scalar. This leads to the corresponding result concerning linear mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one. We also discuss some consequences of this characterization theorem.