The polar decomposition for adjointable operators on Hilbert $C^{\ast }$-modules and centered operators

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