Spectrum of a Linear Operator

[Nusc]

Homework Statement

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 $$\sigma_p(M_\phi)$$

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

The Attempt at a Solution

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

 

[mighty2000]

First, let's define the operator M_\phi on L^p(\mu) more explicitly. For any f \in L^p(\mu) , we have:
M_\phi f = \phi f = \phi(x)f(x) d\mu(x)
where \phi(x) is the pointwise multiplication of \phi and f and d\mu(x) is the measure on (X,\Omega) .

Now, let's consider the spectrum of the operator M_\phi . The spectrum, denoted by \sigma(M_\phi) , is the set of all complex numbers \lambda such that M_\phi - \lambda I is not invertible, where I is the identity operator. In other words, \sigma(M_\phi) is the set of all \lambda such that there exists f \in L^p(\mu) such that (M_\phi - \lambda I) f = 0 .

To find the spectrum, we need to solve the equation (M_\phi - \lambda I) f = 0 . We can rewrite this equation as:
\phi(x)f(x) - \lambda f(x) = 0
which implies that:
f(x)(\phi(x) - \lambda) = 0
Since f(x) cannot be zero for all x \in X (otherwise, f would not be a non-zero function in L^p(\mu) ), we must have:
\phi(x) - \lambda = 0
which gives us:
\phi(x) = \lambda
Therefore, the spectrum of M_\phi is given by:
\sigma(M_\phi) = \{\lambda \in \mathbb{C}: \phi(x) = \lambda \text{ for almost all } x \in X \}

Next, let's consider the approximate point spectrum of M_\phi . The approximate point spectrum, denoted by \sigma_{ap}(M_\phi) , is the set of all complex numbers \lambda such that there exists a sequence \{f_n\} \subset L^p(\mu) with \|f_n\| = 1 for all n and \|(M_\phi - \