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Math Help - Spectrum - proof of the theorem

  1. #1
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    Spectrum - proof of the theorem

    Hi,

    Where I can find (in which book) proof of the following theorem:

    "The spectrum of a function in the algebra is the set of values of the function"

    or how can I proof it?

    Thanks for any help.
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  2. #2
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    Quote Originally Posted by Arczi1984 View Post
    Hi,

    Where I can find (in which book) proof of the following theorem:

    "The spectrum of a function in the algebra is the set of values of the function"

    or how can I proof it?
    If C(S) is the Banach algebra of continuous functions on a compact set S, and f\in C(S), the inverse of f-\lambda1 will be the function g given by g(t) = \frac1{f(t)-\lambda} (provided that such a function exists). If f(t) is never equal to \lambda then g will be defined on all of S, bounded (because S is compact) and continuous. Therefore f-\lambda1 has an inverse in C(S) and hence \lambda\notin\sigma(f). But if there is a value of t for which f(t)=\lambda then g is not defined at that value of t. In that case f-\lambda1 has no inverse in C(S) and hence \lambda\in\sigma(f).
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