Spectrum (Operator)

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

$$\rho(T) = \{\lambda: (T - \lambda I){\rm\ is\ invertible}\},$$

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

See also Hilbert Space