Normality of adjointable module maps

Abstract

Normality of bounded and unbounded adjointable operators are discussed. Suppose $T$ is an adjointable operator between Hilbert C*-modules which has polar decomposition, then $T$ is normal if and only if there exists a unitary operator $ \mathcal{U}$ which commutes with $T$ and $T^*$ such that $T=\mathcal{U} \, T^*.$ Kaplansky's theorem for normality of the product of bounded operators is also reformulated in the framework of Hilbert C*-modules.