Let be a linear operator on a separable 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.
If is a domain in (i.e., a Lebesgue measurable subset of with finite nonzero Lebesgue measure), then a set is a spectrum of if is an orthogonal basis of (Iosevich et al. 1999).