I need to show that if is an operator defined by and is NOT in the set of eigenvalues of then is invertible. I also need to find the set of approximate eigenvalues of and the spectrum of .
So far I have argued that since is not a eigenvalue of and is not zero, there is no vector such that which implies that and therefore and therefore hence is non singular and therefore invertible. Is this correct?
No, it's not. There are two things wrong with it. First, on an infinite-dimensional space like , an operator can be injective without being surjective. So the fact that for all does not imply that is invertible. Second, I'm assuming that you are dealing with bounded operators here. So it's not enough to show that has an algebraic inverse, you need to show that the inverse is a bounded operator.