Spectrum (functional Analysis) - Spectrum of A Bounded Operator

Spectrum of A Bounded Operator

The spectrum of a bounded linear operator T acting on a Banach space X is the set of complex numbers λ such that λI − T does not have an inverse that is a bounded linear operator. If λI − T is invertible then that inverse is linear (this follows immediately from the linearity of λI − T), and by the bounded inverse theorem is bounded. Therefore the spectrum consists precisely of those λ where λI − T is not bijective.

The spectrum of a given operator T is denoted σ(T), and the resolvent set (the set of points not in the spectrum) is denoted ρ(T).