• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Spectrum of a Linear Operator

  • Thread starter Nusc
  • Start date
754
2
1. Homework Statement

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. Homework 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.
 

Want to reply to this thread?

"Spectrum of a Linear Operator" You must log in or register to reply here.

Related Threads for: Spectrum of a Linear Operator

  • Posted
Replies
12
Views
2K
Replies
0
Views
1K
Replies
0
Views
658
Replies
3
Views
3K
  • Posted
Replies
1
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top