On Power Finite Rank Operators

Key Word: power finite rank operator   Drazin invertible   eventual topological uniform descent   Riesz operator
Fund ProjectL:Supported by the Talents Cultivation Program for Outstanding Youth Scientists in Fujian Universities (Grant Nos.Min Education [2015] 54 and [2016] 23), the National Natural Science Foundation of China (Grant No.11401097) and the Natural Science Foundation of Fujian Province (Grant No.2016J05001).
 Author Name Affiliation Qingping ZENG College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fujian 350002, P. R. China Zhenying WU College of Mathematics and Informatics, Fujian Normal University, Fujian 350117, P. R. China
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An operator $F \in \mathcal{B}(X)$ is called power finite rank if $F^{n}$ is of finite rank for some $n \in \mathbb{N}$. In this note, we provide several interesting characterizations of power finite rank operators. In particular, we show that the class of power finite rank operators is the intersection of the class of Riesz operators and the class of operators with eventual topological uniform descent.