Topological divisor of zero explained

In mathematics, an element

z of a Banach algebra A is called a topological divisor of zero if there exists a sequence x _{1,x} _{2,x} _{3,...} of elements of A such that

1. The sequence

zx _{n} converges to the zero element, but

1. The sequence

x _{n} does not converge to the zero element.If such a sequence exists, then one may assume that \left \Vert x _{n} \right \| = 1 for all n .

If

A is not commutative, then z is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly.

Examples

• If

A has a unit element, then the invertible elements of A form an open subset of A , while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero.

• In particular, any quasinilpotent element is a topological divisor 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.

Generalization

The notion of a topological divisor of zero may be generalized to any topological algebra. If the algebra in question is not first-countable, one must substitute nets for the sequences used in the definition.