Self Adjoint Operators On Hilbert Space
Hilbert spaces are Banach spaces, so the above discussion applies to bounded operators on Hilbert spaces as well, although possible differences may arise from the adjoint operation on operators. For example, let H be a Hilbert space and T ∈ L(H), σ(T*) is not σ(T) but rather its image under complex conjugation.
For a self adjoint T ∈ L(H), the Borel functional calculus gives additional ways to break up the spectrum naturally.
Read more about this topic: Decomposition Of Spectrum (functional Analysis)
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