1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Spectrum of a Linear Operator

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

    [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
    [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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted