Gibbons–Hawking–York boundary term: Difference between revisions

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A '''Neumann series''' is a [[series (mathematics)|mathematical series]] of the form
:<math> \sum_{k=0}^\infty T^k </math>
where ''T'' is an [[Operator (mathematics)|operator]]. Hence, ''T<sup>k</sup>'' 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 equation]]s. It is also important when studying the [[spectrum (functional analysis)|spectrum]] of bounded operators.
 
== Properties ==
 
Suppose that ''T'' is a bounded operator on the [[normed vector space]] ''X''. If the Neumann series [[Convergent series|converges]] in the [[operator norm]], then Id – ''T'' is [[Invertible matrix|invertible]] and its inverse is the series:
:<math> (\mathrm{Id} - T)^{-1} = \sum_{k=0}^\infty T^k </math>,
where <math> \mathrm{Id} </math> is the [[identity operator]] in ''X''. To see why, consider the partial sums
:<math>S_n := \sum_{k=0}^n T^k</math>.
Then we have
:<math>\lim_{n \rightarrow \infty}(\mathrm{Id}-T)S_n = \lim_{n \rightarrow \infty}\left(\sum_{k=0}^n T^k - \sum_{k=0}^n T^{k+1}\right) = \lim_{n \rightarrow \infty}\left(\mathrm{Id} - T^{n+1}\right) = \mathrm{Id}.</math>
 
One case in which convergence is guaranteed is when ''X'' is a [[Banach space]] and |''T''| < 1 in the operator norm. However, there are also results which give weaker conditions under which the series converges.
 
== 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'' &rarr; ''B''<nowiki>'</nowiki> be an invertible operator and let ''T'': ''B'' &rarr; ''B''<nowiki>'</nowiki> be another operator. If |''S'' – ''T'' | < |''S''<sup>–1</sup>|<sup>–1</sup>, then ''T'' is also invertible. This follows by writing ''T'' as
:<math> T = S ( \mathrm{Id} - (\mathrm{Id} - S^{-1} T ))\, </math>
and applying the result in the previous section on the second factor. The norm of ''T''<sup>–1</sup> can be bounded by
:<math> |T^{-1}| \le \tfrac{1}{1-q} |S^{-1}| \quad\text{where}\quad q = |S-T| \, |S^{-1}|. </math>
 
== References ==
* {{cite book| last=Werner| first=Dirk| year=2005| title=Funktionalanalysis | language=German| publisher=Springer Verlag| isbn=3-540-43586-7}}
 
{{iw-ref|de|Neumann-Reihe}}
 
[[Category:Functional analysis]]
[[Category:Mathematical series]]

Revision as of 06:00, 12 October 2013

A Neumann series is a mathematical series of the form

∑k=0∞Tk

where T is an operator. Hence, Tk 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.

Properties

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:

(Id−T)−1=∑k=0∞Tk,

where Id is the identity operator in X. To see why, consider the partial sums

Sn:=∑k=0nTk.

Then we have

limn→∞(Id−T)Sn=limn→∞(∑k=0nTk−∑k=0nTk+1)=limn→∞(Id−Tn+1)=Id.

One case in which convergence is guaranteed is when X is a Banach space and |T| < 1 in the operator norm. However, there are also results which give weaker conditions under which the series converges.

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. This follows by writing T as

T=S(Id−(Id−S−1T))

and applying the result in the previous section on the second factor. The norm of T–1 can be bounded by

|T−1|≤11−q|S−1|whereq=|S−T||S−1|.

References

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