Let

$\displaystyle 1 \leq p \leq \infty $ and let $\displaystyle (X,\Omega, \mu) $

be a $\displaystyle \sigma$-finite measure space.

For $\displaystyle \phi \in L^\infty(\mu) $, define $\displaystyle M_\phi $ on $\displaystyle L^p(\mu) $ by $\displaystyle M_\phi f = \phi f \forall f \in L^(X,\Omega,\mu)$.

I need to find the following:

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

where

$\displaystyle \sigma(M_\alpha) = \{\alpha \in F: M_\alpha-\alpha $ is not invertible $\displaystyle \} $

$\displaystyle \sigma_{ap} \equiv \{ \lambda \in C$ : there is a sequence $\displaystyle \{x_n\} $ in X such that $\displaystyle ||x_n|| = 1 $for all n and $\displaystyle ||(A-\lambda)x_n||\rightarrow 0 \}$ and

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

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