topologically nilpotent
An element $a$ in a normed ring^{} $A$ is said to be topologically nilpotent if
$$\underset{n\to \mathrm{\infty}}{lim}{\parallel {a}^{n}\parallel}^{\frac{1}{n}}=0.$$ 
Topologically nilpotent elements are also called quasinilpotent.
Remarks.

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Any nilpotent element^{} is topologically nilpotent.

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If $a$ and $b$ are topologically nilpotent and $ab=ba$, then $ab$ is topologically nilpotent.

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When $A$ is a unital Banach algebra^{}, an element $a\in A$ is topologically nilpotent iff its spectrum $\sigma (a)$ equals $\{0\}$.
Title  topologically nilpotent 

Canonical name  TopologicallyNilpotent 
Date of creation  20130322 16:12:04 
Last modified on  20130322 16:12:04 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 46H05 
Synonym  quasinilpotent 