spectrum is a non-empty compact set

Theorem - Let 𝒜 be a complex Banach algebraMathworldPlanetmath with identity element. The spectrum of each a𝒜 is a non-empty compact set in .

Remark : For Banach algebras over the spectrum of an element is also a compact set, although it can be empty. To assure that it is not the empty setMathworldPlanetmath, proofs usually involve Liouville’s theorem (http://planetmath.org/LiouvillesTheorem2) for of a complex with values in a Banach algebra.

Proof : Let e be the identity element of 𝒜. Let σ(a) denote the spectrum of the element a𝒜.

  • - For each λ such that |λ|>a one has λ-1a<1, and so, by the Neumann series (http://planetmath.org/NeumannSeriesInBanachAlgebras), e-λ-1a is invertible. Since


    we see that a-λe is also invertible.

    We conclude that σ(a) is contained in a disk of radius a, and therefore it is boundedPlanetmathPlanetmathPlanetmathPlanetmath.

    Let ϕ:𝒜 be the function defined by


    It is known that the set 𝒢 of the invertible elements of 𝒜 is open (see this entry (http://planetmath.org/InvertibleElementsInABanachAlgebraFormAnOpenSet)).

    Since ϕ-1(𝒢)=-σ(a) and ϕ is a continuous functionMathworldPlanetmathPlanetmath we see that that σ(a) is a closed setPlanetmathPlanetmath in .

    As σ(a) is a bounded closed subset of , it is compact.

  • Non-emptiness - Suppose that σ(a) was empty. Then the resolvent Ra is defined in .

    We can see that Ra is bounded since it is continuous in the closed disk |λ|<a and, for λ>a, we have (again, by the Neumann series (http://planetmath.org/NeumannSeriesInBanachAlgebras))

    Ra(λ) = (a-λe)-1
    = λ-1(e-λ-1a)-1
    = 1|λ|-a

    and therefore lim|λ|Ra(λ)=0, which shows that Ra is bounded.

    The resolvent function, Ra, is analyticPlanetmathPlanetmath (http://planetmath.org/BanachSpaceValuedAnalyticFunctions) (see this entry (http://planetmath.org/ResolventFunctionIsAnalytic)). As it is defined in , it is a bounded entire function. Applying Liouville’s theorem (http://planetmath.org/LiouvillesTheorem2) we conclude that it must be constant (see this this entry (http://planetmath.org/BanachSpaceValuedAnalyticFunctions) for an idea of how holds for Banach spaceMathworldPlanetmath valued functions).

    Since Ra(λ) convergesPlanetmathPlanetmath to 0 as |λ| we see that Ra must be identically zero.

    Thus, we have arrived to a contradictionMathworldPlanetmathPlanetmath since 0 is not invertible.

    Therefore σ(a) is non-empty.

Title spectrum is a non-empty compact set
Canonical name SpectrumIsANonemptyCompactSet
Date of creation 2013-03-22 17:25:05
Last modified on 2013-03-22 17:25:05
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Theorem
Classification msc 46H05