Restriction of an operator to the range of its powers

Let T be a bounded linear operator acting on a Banach space X. For each integer n, define ${\displaystyle T_n}$ to be the restriction of T to ${\displaystyle R\left(T^n\right)}$ viewed as a map from ${\displaystyle R\left(T^n\right)}$ into ${\displaystyle R\left(T^n\right)}$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator ${\displaystyle T_n}$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where ${\displaystyle T_n}$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological uniform descent.