Neumann series

A Neumann series is a mathematical series of the form

${\displaystyle \sum _{k=0}^{\infty }T^{k}}$

where T is an operator and ${\displaystyle T^{k}:={}T^{k-1}\circ {T}}$ its k times repeated application. 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.

Properties

Suppose that T is a bounded linear 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:

${\displaystyle (\mathrm {Id} -T)^{-1}=\sum _{k=0}^{\infty }T^{k}}$,

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

${\displaystyle S_{n}:=\sum _{k=0}^{n}T^{k}}$.

Then we have

${\displaystyle \lim _{n\rightarrow \infty }(\mathrm {Id} -T)S_{n}=\lim _{n\rightarrow \infty }\left(\sum _{k=0}^{n}T^{k}-\sum _{k=1}^{n}T^{k+1}\right)=\lim _{n\rightarrow \infty }\left(\mathrm {Id} -T^{n+1}\right)=\mathrm {Id} .}$

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

${\displaystyle (1-x)\cdot (1+x+x^{2}+\cdots +x^{n-1}+x^{n})=1-x^{n+1},}$

${\displaystyle 1+x+x^{2}+\cdots ={\frac {1}{1-x}}.}$

One case in which convergence is guaranteed is when X is a Banach space and |T| < 1 in the operator norm or ${\displaystyle \sum |T^{n}|}$ is convergent. However, there are also results which give weaker conditions under which the series converges.

Example

Let ${\displaystyle C\in \mathbb {R} ^{3\times 3}}$ be given by:

${\displaystyle {\begin{pmatrix}0&{\frac {1}{2}}&{\frac {1}{4}}\\{\frac {5}{7}}&0&{\frac {1}{7}}\\{\frac {3}{10}}&{\frac {3}{5}}&0\end{pmatrix}}.}$

We need to show that C is smaller than unity in some norm. Therefore, we calculate:

${\displaystyle {\begin{aligned}||C||_{\infty }&=\max _{i}\sum _{j}|c_{ij}|=\max \left\lbrace {\frac {3}{4}},{\frac {6}{7}},{\frac {9}{10}}\right\rbrace ={\frac {9}{10}}<1.\end{aligned}}}$

Thus, we know from the statement above that ${\displaystyle (I-C)^{-1}}$ exists.

The set of invertible operators is open

A corollary is that the set of invertible operators between two Banach spaces B and B' is open in the topology induced by the operator norm. Indeed, let S : B → B' be an invertible operator and let T: B → B' be another operator. If

|S – T | < |S⁻¹|⁻¹,

then T is also invertible.

Since |Id – S⁻¹T| < 1, the Neumann series Σ(Id – (S⁻¹T))^k is convergent. Therefore, we have

T⁻¹S = (Id – (Id – S⁻¹T))⁻¹ = Σ(Id – (S⁻¹T))^k.

Taking the norms, we get |T⁻¹S| ≤ 1/(1 – |Id – (S⁻¹T)|).

The norm of T⁻¹ can be bounded by

${\displaystyle |T^{-1}|\leq {\tfrac {1}{1-q}}|S^{-1}|\quad {\text{where}}\quad q=|S-T|\,|S^{-1}|.}$

References

• Werner, Dirk (2005). Funktionalanalysis (in German). Springer Verlag. ISBN 3-540-43586-7.