Linear preserver problems in matrix positivity theory

Abstract

Linear preserver problems have been a central focus of research in matrix theory and operator theory for more than a century, beginning with Frobenius' 1897 characterization of determinant preserving linear maps on the space of complex matrices. Since this foundational result, considerable work has examined linear preservers of diverse subsets, functions, and relations across different matrix and operator spaces. The purpose of this survey is to present the current state of research on linear preserver problems for several positivity classes of matrices. We provide an overview of recent developments in the literature and, for each positivity class considered, identify gaps that remain to guide future research.