The $$(n+1)$$-centered operator on a Hilbert $$C^*$$-module

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