From Chapter 8, only the material covered in the lectures is included.

This translates to roughly the following from the textbook:

All of Sec 8.1.

All of Sec 8.2 (except example 8.13).

From Sec 8.1: Everything up until (and including) Theorem 8.18.
Fredholm operators are only included to the extent that they were
discussed in class. You do not need to memorize the definition of a
Fredholm operator, or the definition of the index of an operator.  You
should know that an operator of the form I+K where K is compact has
closed range and finite dimensional nullspace. Then the fact that it
satisfies the Fredholm alternative is a direct consequence of Theorem
8.17.

Sec 8.5 is omitted in its entirety.

In Sec 8.6, theorem 8.40 and example 8.41 are important.  You should
have some idea of how they follow from theorem 8.39, but you do not
need to memorize theorem 8.39, its proof, or the proof of theorem
8.40. The examples are very useful.  Prop 8.44 is included (with
proof). Theorem 8.45 is included, but its proof is not. All material
after Theorem 8.45 is omitted.

In Chapter 9, all material up to (and including) Corollary 9.14 
is included, except the proof of Prop 9.7. You should know that
there are many tools from complex analysis that can be applied to 
resolvent operators (this is an important part of spectral theory)
but this is not a core part of the course.