# 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 $\{0\}$.

Title topologically nilpotent TopologicallyNilpotent 2013-03-22 16:12:04 2013-03-22 16:12:04 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 46H05 quasinilpotent