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Zbl 0978.47011
Berkani, M.
Restriction of an operator to the range of its powers. (English)
[J] Stud. Math. 140, No.2, 163-175 (2000). ISSN 0039-3223; ISSN 1730-6337/e

Let $T$ be a bounded linear operator on a Banach space $X$. The author investigates conditions render which the restriction of $T^h$ on $R(T^n)$ has some regularity properties. Among the others the following results are proved:\par Theorem 1. Let $T\in L(X)$. Then $R(T^{\delta(T)})$ is closed and $\delta(T):= \inf\{n: R(T^n)= R(T^{n+1})\}< \infty$ iff $\text{dis}(T):= \inf\{n\in\bbfN: \forall m\in\bbfN: m\ge n\Rightarrow R(T^n)\cap{\cal N}(T)\subseteq R(T^m)\cap{\cal N}(T)\}= d$ and\par a) $R(T^m)$ is a closed subspace of $X$ for any $n\ge d$,\par b) $R(T)+{\cal N}(T^d)$ is a closed subspace of $X$.\par Theorem 2. Let $\text{dis}(T)= d$. Then the range $R(T^{d+1})$ is closed and $\text{dis}(T)= d$ iff $T$ is quasi-Fredholm operator.
[S.I.Piskarev (Moskva)]

MSC 2000:: *47A53 (Semi-)Fredholm operators; index theories
47A55 Perturbation theory of linear operators

Keywords: range of powers; regularity; quasi-Fredholm operator

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