Abstract
In this purely mathematically-flavoured chapter we introduce the basic spectral theory for operators on normed spaces, up to the notion of spectral measure and the spectral decomposition theorem for normal operators in B(H), with H a Hilbert space. (The spectral theorem for unbounded self-adjoint operators will be discussed in the next chapter.) Here we present a number of general result in the abstract theory of C*-algebras and *-homomorphisms.
A mathematician plays a game and invents the rules. A physicist plays a game whose rules are dictated by Nature. As time goes by it is more and more evident that the rules the mathematician finds appealing are precisely those Nature has chosen.
P.A.M. Dirac
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Notes
- 1.
The analogous property for normal operators (hence unitary or self-adjoint too) in 𝔅(H) is contained in Proposition 3.54(b).
- 2.
2 For any z ∈ D ε we can choose an open disc, centred at z of positive radius δ < ε, so that D δ ⊂ Dε . This gives an open covering of Dε . Lindelöf’s lemma (Theorem 1.8) guarantees
we can extract a countable subcovering \({\left\{{D}_{\delta i}^{(i)}\right\}}_{i\in \mathbb{N}}.\)
Since \({D}_{\epsilon }={\cup}_{i\in \mathbb{N}}{D}_{\delta i}^{(i)}\) then, \({\mu}_{\beta}\left({E}_{\epsilon}\right)\le {\displaystyle {\sum}_{i\in \mathbb{N}{\mu}_{\beta }}\left({D}_{\delta_i}^{(i)}\right).}\) If we had \({\mu}_{\beta}\left({D}_{\delta_i}^{(i)}\right)=0\) for any i, we would obtain μ β (D ε ) = 0.
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Moretti, V. (2013). Spectral Theory I: generalities, abstract C *-algebras and operators in B(H). In: Spectral Theory and Quantum Mechanics. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2835-7_8
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DOI: https://doi.org/10.1007/978-88-470-2835-7_8
Publisher Name: Springer, Milano
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