Neumann series

A Neumann series is a mathematical series of the form

where T is an operator. Hence, T^k is a mathematical notation for k consecutive operations of the operator T. This generalizes the geometric series.

The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis. It forms the basis of the Liouville-Neumann series, which is used to solve Fredholm integral equations. It is also important when studying the spectrum of bounded operators.

Suppose that T is a bounded operator on the normed vector space X. If the Neumann series converges in the operator norm, then Id – T is invertible and its inverse is the series:

where ${\displaystyle \mathrm {Id} }$ is the identity operator in X. To see why, consider the partial sums

Then we have

This result on operators is analogous to geometric series in ${\displaystyle \mathbb {R} }$, in which we find that:

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