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

Let $n$ be any natural number. The $n$-centered operator is introduced for adjointable operators on Hilbert $C^{\ast }$-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$, if $T$ is Moore-Penrose invertible and is $n$-centered, then its Moore-Penrose inverse is also $n$-centered. A Hilbert space operator $T$ is constructed such that $T$ is $n$-centered, whereas it fails to be $(n+1)$-centered.