spectral mapping theorem

Let 𝒜 be a unital C*-algebra (http://planetmath.org/CAlgebra). Let x be a normal element in 𝒜 and σ(x) be its spectrum.

The continuous functional calculus provides a C*-isomorphismPlanetmathPlanetmathPlanetmathPlanetmath



between the C*-algebra C(σ(x)) of complex valued continuous functionsMathworldPlanetmathPlanetmath on σ(x) and the C*-subalgebra 𝒜[x]𝒜 generated by x and the identity of 𝒜.

Spectral Mapping Theorem - Let x𝒜 be as above. Let fC(σ(x)). Then


Proof : Since C(σ(x)) and 𝒜[x] are isomorphic we must have


where σ𝒜[x](f(x)) denotes the spectrum of f(x) relative to the subalgebra 𝒜[x].

By the spectral invariance theorem we have σ𝒜[x](f(x))=σ(f(x)). Hence


Thus, we only have to prove that f(σ(x))=σ(f).

f is defined on σ(x) so f(σ(x)) is precisely the image of f.

Let λ. The function f-λ is invertible if and only if f-λ has no zeros.

Equivalently, f-λ is not invertible if and only if f-λ has a zero, i.e. f(λ0)=λ for some λ0.

The previous statement can be reformulated as: λσ(f) if and only if λ is in the image of f.

We conclude that σ(f)=f(σ(x)), and this proves the theorem.

Title spectral mapping theorem
Canonical name SpectralMappingTheorem
Date of creation 2013-03-22 17:30:08
Last modified on 2013-03-22 17:30:08
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 46L05
Classification msc 47A60
Classification msc 46H30