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).
Read more about this topic: Spectrum (functional Analysis)
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