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.

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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 $a b$ 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 $\{0\}$ .