  Let $n$ be any natural number. The $n$-centered operator is introduced for
adjointable operators on Hilbert $C^*$-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.
