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

Abstract

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.

arXiv commentA minor revision is made. Accepted for publication in Advances in Operator Theory