Topological divisor of zero

In mathematics, an element z of a Banach algebra A is called a topological divisor of zero if there exists a sequence x1, x2, x3, ... of elements of A such that

  1. The sequence zxn converges to the zero element, but
  2. The sequence xn does not converge to the zero element.

If such a sequence exists, then one may assume that ||xn|| = 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 , which is injective, not surjective, but whose image is dense in , 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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.