Neumann series

In mathematics , a Neumann series (or Neumann series ) is a series of the form , where is a continuous linear operator on a normalized space and . ${\displaystyle \textstyle \sum _{n=0}^{\infty }T^{n}}$ ${\displaystyle T:X\rightarrow X}$ ${\displaystyle \left.X\right.}$ ${\displaystyle \left.T\right.^{0}:=\mathrm {Id} _{X}}$

The series corresponds formally to a geometric series and is named after the mathematician Carl Gottfried Neumann , who used it in 1877 in potential theory. She finds u. a. Use in functional analysis to solve operator equations and is important when investigating continuous operators, cf. Spectrum (operator theory) .

properties

Be a normed space and a compact operator, . It is , thus steady - - the space of linear bounded operators on . ${\displaystyle \left(X,\left\|\cdot \right\|\right)}$ ${\displaystyle T\colon X\rightarrow X}$ ${\displaystyle T\in L(X)}$ ${\displaystyle L(X)}$ ${\displaystyle X}$

• If the Neumann series converges in space with respect to the operator norm , then is invertible and it holds ${\displaystyle \sum _{n=0}^{\infty }T^{n}}$ ${\displaystyle \left.L(X)\right.}$ ${\displaystyle A=\left(\mathrm {Id} -T\right)}$

${\displaystyle A^{-1}=\left(\mathrm {Id} -T\right)^{-1}=\sum \limits _{k=0}^{\infty }T^{k}}$.

• The Neumann series converges if is a Banach space and holds for the operator norm . Then also applies: ${\displaystyle \left(X,\left\|.\right\|\right)}$ ${\displaystyle \left\|T\right\|<1}$

${\displaystyle \left\|(\mathrm {Id} -T)^{-1}\right\|\leq \left(1-\|T\|\right)^{-1}}$.

• Weaker conditions are also known under which the series can converge, e.g. B. it is sufficient if the condition is only valid for a power of the operator . Then ${\displaystyle T}$ ${\displaystyle \;\|T^{n}\|<1\;}$

${\displaystyle {\begin{aligned}(I-T)^{-1}=&\left(I+T+T^{2}+\dots +T^{n-1}\right)\cdot \left(I-T^{n}\right)^{-1}\\=&\left(I+T+T^{2}+\dots +T^{n-1}\right)\cdot \sum \limits _{k=0}^{\infty }T^{kn}.\end{aligned}}}$

Invertibility of linear operators

Is a Banach space, e.g. B. , and a bounded operator, e.g. B. a square matrix , then for each scaling factor as ${\displaystyle V}$ ${\displaystyle V=\mathbb {R} ^{n}}$ ${\displaystyle A\colon V\to V}$ ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle A}$ ${\displaystyle \gamma >0}$

${\displaystyle A={\tfrac {1}{\gamma }}(I-T_{\gamma })\;}$ With ${\displaystyle \;T_{\gamma }=I-\gamma \,A}$

being represented.

If there is now a scaling factor with which the induced operator norm applies, then it is invertible and the inverse is, using the Neumann series, ${\displaystyle \|T_{\gamma }\|_{{}_{V\to V}}<1}$ ${\displaystyle A}$

${\displaystyle A^{-1}=\gamma \left(I+\sum _{k=1}^{\infty }T_{\gamma }{}^{k}\right)=\gamma \left(I+\sum _{k=1}^{\infty }(I-\gamma \,A)^{k}\right)}$.

Openness of the set of invertible operators

Let be two Banach spaces and an invertible operator. Then for every further operator : ${\displaystyle B,B'}$ ${\displaystyle S\colon B\to B'}$ ${\displaystyle T\colon B\to B'}$

If the estimate with applies to the distance in the operator norm from to , then it is also invertible and the inverse has the operator norm ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle \|T-S\|\leq q\,\|S^{-1}\|^{-1}}$ ${\displaystyle 0<q<1}$ ${\displaystyle T}$

${\displaystyle \|T^{-1}\|\leq {\tfrac {1}{1-q}}\|S^{-1}\|}$.

As proof: It is broken down and the Neumann series is applied to the second factor. Convergence is assured, because according to the prerequisite: ${\displaystyle T=S(I-(I-S^{-1}T))}$

${\displaystyle \|I-S^{-1}T\|\leq \|S^{-1}\|\,\|S-T\|\leq q<1}$.

As a result, the set of invertible operators is open with regard to the topology of the operator norm.

literature

• Dirk Werner : Functional Analysis . Springer Verlag, 2005. ISBN 3-540-43586-7