The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators

Abstract

‎‎‎Let $T$ be an adjointable operator between two Hilbert $C^*$-modules‎, ‎and let $T^*$ be the adjoint operator of $T$‎. ‎The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac{1}{2}$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$‎, ‎where $U$ is a partial isometry‎, ‎$\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$‎, ‎respectively‎. ‎Based on this new characterization of the polar decomposition‎, ‎an application to the study of centered operators is carried out‎.