resolvent function is analytic

Theorem - Let 𝒜 be a complex Banach algebraMathworldPlanetmath with identity elementMathworldPlanetmath e. Let x𝒜 and σ(x) denote its spectrum.

Then, the resolvent function ( Rx:-σ(x)𝒜 defined by Rx(λ)=(x-λe)-1 is analytic (

Moreover, for each λ0-σ(x) it has the power series

Rx(λ)=n=0Rx(λ0)n+1(λ-λ0)n (1)

where the series converges absolutely for each λ in the open disk centered in λ0 given by

|λ-λ0|<1Rx(λ0) (2)

Proof : Analyticity is defined for functions whose domain is open.

Thus, we start by proving that -σ(x) is an open set in . To do so it is enough to prove that for every λ0-σ(x) the open disk defined by (2) above is contained in -σ(x).

Let λ0-σ(x) and λ be such that


Then (λ-λ0)Rx(λ0)<1 and by the Neumann series ( e-(λ-λ0)Rx(λ0) is invertible.

Since λ0σ(x) it follows that (x-λ0e) is invertible.

Hence, from the equality

x-λe=x-λ0e-(λ-λ0)e=(x-λ0e)[e-(λ-λ0)Rx(λ0)] (3)

we conclude that x-λe is also invertible, i.e. λ-σ(x). Thus -σ(x) is open.

The above proof also pointed out that for every λ0, Rx is defined in the open disk of radius 1Rx(λ0) centered in λ0.

We now prove the analyticity of the .

Taking inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on the equality (3) above one obtains


Again, by the Neumann series (, one obtains

Title resolvent function is analytic
Canonical name ResolventFunctionIsAnalytic
Date of creation 2013-03-22 17:29:36
Last modified on 2013-03-22 17:29:36
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 8
Author asteroid (17536)
Entry type Theorem
Classification msc 46H05
Classification msc 47A10