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.

• 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 $\{0\}$.