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Homework Help: Spectrum of a Linear Operator

  1. Mar 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Let
    [tex] 1 \leq p \leq \infty [/tex] and let [tex] (X,\Omega, \mu) [/tex]
    be a [tex]\sigma[/tex]-finite measure space.

    For [tex]\phi \in L^\infty(\mu) [/tex], define [tex] M_\phi [/tex] on [tex] L^p(\mu) [/tex] by [tex] M_\phi f = \phi f \forall f \in L^(X,\Omega,\mu)[/tex].
    I need to find the following:

    [tex] \sigma(M_\phi) [/tex], [tex]\sigma_ap(M_\phi)[/tex], and [tex]\sigma_p(M_\phi)[/tex]





    2. Relevant equations
    where
    [tex] \sigma = \{\alpha \in F: a-\alpha [/tex] is not invertible [tex]\} [/tex]
    [tex]\sigma_{ap} \equiv \{ \lambda \in C[/tex] : there is a sequence [tex]\{x_n\} [/tex] in X such that [tex]||x_n|| = 1 [/tex] for all n and [tex] ||(A-\lambda)x_n||\rightarrow 0 \}[/tex] and

    and [tex]\sigma_p \equiv \{\lambda in C: ker(A-\lambda) \neq (0)\}[/tex]


    3. The attempt at a solution

    Your help is invaluable - I have no idea how to do this.
     
  2. jcsd
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