## Restriction of an operator to the range of its powers

### Volume 140 / 2000

Studia Mathematica 140 (2000), 163-175
DOI: 10.4064/sm-140-2-163-175

#### 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.