The proof of 1. is a simple verification. First, show that

is a bounded operator on

. Then check that

and

.

For 2., you should show that the spectrum of

is the range of f,

This follows fairly easily from 1., because the function

is invertible provided that it never vanishes. The point spectrum of

, on the other hand, is the set of points

such that the function f takes the constant value

on some interval of positive length.

For 3., I think I would use the fact that the closure of the image of a bounded operator is the orthogonal complement of the kernel of the adjoint. So try to show that