Let T be a linear operator on a separable Hilbert space. The spectrum σ(T) of T is the set of λ such that (T - λ I) is not invertible on all of the Hilbert space, where the λs are complex numbers and I is the identity operator. The definition can also be stated in terms of the resolvent of an operator ρ(T) = {λ:(T - λ I) is invertible}, and then the spectrum is defined to be the complement of ρ(T) in the complex plane. It is easy to demonstrate that ρ(T) is an open set, which shows that the spectrum σ(T) is closed.
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