Topological divisor of zero

In mathematics, in a topological algebra "A", $zin\; A$ is a topological divisor of zero if there exists a neighbourhood "U" of zero and a net $(x\_i)\_\{iin\; I\}$ with $forall\; iin\; I,\; x\_i\; in\; Asetminus\; U$ and $zx\_i\; longrightarrow\; Oin\; A.$ If the topological algebra is not commutative use left resp. right topological divisor of zero.

They are not invertible, otherwise multiplying by the inverse would give $x\_ilongrightarrow\; 0$, contradicting $x\_iin\; Asetminus\; U$.

Example

In a Banach algebra $(A,|cdot|)$ with a norm $|cdot|$ an element "z" is a topological divisor of zero if and only if it there exists a sequence $(x\_n)$ in "A" such that $|x\_n|=1$ for all $n$ while $lim\_\{n\; ightarrow\; infty\}\; |zcdot\; x\_n|\; =\; 0$

An element of a Banach algebra with unity, which is at the boundary of the closed set of non-invertible elements and the open set of invertible ones, is a left- and right topological divisor of zero. Thus, quasinilpotents are topological divisors of zero (e.g. the Volterra operator).

An operator on a Banach space $X$, which is injective, not surjective, but whose image is dense in $X$, is a left topological divisor of zero.