# Homework Help: Spectrum of a Linear Operator

1. Mar 23, 2009

### Nusc

1. The problem statement, all variables and given/known data

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

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

$$\sigma(M_\phi)$$, $$\sigma_ap(M_\phi)$$, and $$\sigma_p(M_\phi)$$

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

and $$\sigma_p \equiv \{\lambda in C: ker(A-\lambda) \neq (0)\}$$

3. The attempt at a solution

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