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Abstract

Let T be a bounded linear operator acting on a Banach space X. For each integer n, define $T_n$ to be the restriction of T to $ R(T^n) $ viewed as a map from $R(T^n)$ into $R(T^n)$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator $T_n$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where $T_n$ 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.