## Spectrum (Operator)

Let be an Operator on a Hilbert Space. The spectrum of is the set of such that is not invertible on all of the Hilbert Space, where the s are Complex Numbers and is the Identity Operator. The definition can also be stated in terms of the resolvent of an operator

and then the spectrum is defined to be the complement of in the Complex Plane. It is easy to demonstrate that is an Open Set, which shows that the spectrum is closed (in fact, it is even compact).