  Let $T$ be an adjointable operator on a Hilbert $C^*$-module such that $T$
has the polar decomposition $T=UT|$. For each natural number $n$, $T$ is called
an $(n+1)$-centered operator if $T^k=U^k|T^k|$ is the polar decomposition for
$1\le k\le n+1$. This paper initiates the study of the $(n+1)$-centered
operator via the generalized Aluthge transform and the generalized iterative
Aluthge transform. Some new characterizations of the $(n+1)$-centered operator
are provided.
