Decomposition of the spectrum of a bounded linear operator

1σ(T ) is defined to be the set of λ ∈ C such that v 7→ Tv − λv is not a bijection. It is a fact that if T ∈ B(H) and v 7→ Tv − λv is a bijection then it is an element of B(H). That it is linear can be proved quickly. The fact that it is bounded is proved using the open-mapping theorem, which states that a surjective bounded linear map from one Banach space to another is an open map, from which it follows that a bijective bounded linear map from one Banach space to another has a bounded inverse. 2The point spectrum is often called the discrete spectrum. From the definition by itself it is not apparent what σpoint(T ) has to do either with points or discreteness.