proof of Neumann series in Banach algebras

Let x be an element of a Banach algebraMathworldPlanetmath with identity, x<1. By applying the properties of the Norm in a Banach algebra, we see that the partial sums form a Cauchy sequencePlanetmathPlanetmath: n=lmxnn=lmxn0 for l,m (as is well known from real analysis), so by completeness of the Banach Algebra, the series convergesPlanetmathPlanetmath to some element y=n=0xn.

We observe that for any m,

(1-x)n=0mxn=n=0mxn-n=1m+1xn=1-xm+1 (1)

Furthermore, xm+1xm+1, so limmxm+1=0.

Thus, by taking the limit m on both sides of (1), we get


(We can exchange the limit with the multiplication by (1-x), since the multiplication in Banach algebras is continuousMathworldPlanetmath)

Since the Banach algebra generated by a single element is commutative and (1-x) and y are both in the Banach algebra generated by x, we also get y(1-x)=1. Hence, y=(1-x)-1.

As in the first paragraph, the last claim y11-y again follows by applying the geometric series for real numbers.

Title proof of Neumann series in Banach algebras
Canonical name ProofOfNeumannSeriesInBanachAlgebras
Date of creation 2013-03-22 17:32:40
Last modified on 2013-03-22 17:32:40
Owner FunctorSalad (18100)
Last modified by FunctorSalad (18100)
Numerical id 5
Author FunctorSalad (18100)
Entry type Proof
Classification msc 46H05