Solutions to the operator equation \(f(XX^*)=g(X^*X)\)

Abstract

The characterization of special operators (or matrices), such as normal and self-adjoint (Hermitian) ones, through operator (or matrix) equations has been a topic of significant interest for a long time. In this paper, we explore equations involving products of an operator X and its adjoint \(X^*\). We provide new characterizations of normal and Hermitian matrices using general matrix equations. We then extend and improve these results to operators on Hilbert spaces of any dimension. Moreover, we establish further characterizations of normal and self-adjoint operators (and matrices) based on our findings.