In mathematics, a Neumann series (or Neumann series ), a series of the form, with a continuous linear operator on a normed space and.
The series formally corresponds to a geometric series and is named after the mathematician Carl Gottfried Neumann, who used them in the potential theory 1877. She finds, inter alia, Application in the functional analysis of dissolving operator equations and is important in the study of continuous operators, see spectrum ( operator theory ).
Be a normed space and a compact operator. In this case, the space of the linear limited - and thus steady - by operators of; to write for abbreviation.
- If the Neumann series converges in the space with respect to the operator norm, then is invertible and we have
- The Neumann series converges, if a Banach space and is valid for the operator norm. Then also applies:
- Are weaker conditions known in which the number can be converged, for example, it is sufficient if the condition only applies to a power of the operator. Then
Invertibility of linear operators
If V is a Banach space, for example, and a bounded operator, for example, a square matrix A can of each scaling factor as
There are now a scaling factor, with which the operator applies in the induced norm, then A can be inverted and the inverse is, using the Neumann series,
Openness of the set of invertible operators
Be two Banach spaces and an invertible operator. Then for each additional operator:
As a result, it follows that the set of invertible operators is open with respect to the topology of the operator norm.