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

Any nilpotent element is topologically nilpotent.

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

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