TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Bounded normal operator abstract spectral theorem

Statement

Assume AC. A bounded operator $T$ on a nonzero complex Hilbert space is normal exactly when it is the image of the coordinate function $z \mapsto z$ under a unital star representation of $C (K)$ for some nonempty compact set $K \subseteq C$ ; canonically $K = σ (T)$ and the representation is the continuous functional calculus.

Facts & Assumptions

[A1]

A unital star-homomorphism, or representation, $ρ : C (K) \to B (H)$ is a unital complex-linear multiplicative map with $ρ (\overline{f}) = ρ (f)^{*}$ (C star algebra).

[A2]

For bounded $T$ one has $T^{*} T = T T^{*}$ exactly when $T$ is normal, and the coordinate function $z$ and its conjugate generate the unital $*$ -algebra of functions $q (z, \overline{z})$ (Self-adjoint, positive, unitary and normal operators, C star algebra).

[A3]

For normal $T$ the continuous functional calculus is a unital isometric star-isomorphism $C (σ (T)) \to C^{*} (I, T)$ with $z \mapsto T$ (Continuous functional calculus for bounded normal operators).

[A4]

For every bounded $T$ the spectrum is a nonempty compact subset of $C$ : $B (H)$ is a nonzero unital complex Banach algebra, its algebra spectrum agrees with the operator spectrum by the bounded inverse theorem, and spectra in nonzero unital complex Banach algebras are nonempty and compact (Spectrum and resolvent of a bounded operator, Bounded Hilbert operators form a C star algebra, Bounded inverse theorem, Spectrum is nonempty compact and norm bounded).

[A5]

AC is the hypothesis of the calculus supplier (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ and a bounded operator $T \in B (H)$ .

1.1

Suppose $K \subseteq C$ is nonempty and compact and $T = ρ (z)$ for a unital star-homomorphism $ρ : C (K) \to B (H)$ , where $z$ is the coordinate function; then $T^{*} = ρ (z)^{*} = ρ (\overline{z})$ and $T^{*} T = ρ (\overline{z}) ρ (z) = ρ (\overline{z} z) = ρ (z \overline{z}) = ρ (z) ρ (\overline{z}) = T T^{*}$ , so $T$ is normal.

A1A2

1.2

Conversely, if $T$ is normal, take $K : = σ (T)$ , a nonempty compact Hausdorff space, and the continuous functional calculus $Ψ : C (σ (T)) \to C^{*} (I, T)$ ; it is a unital star-homomorphism into $B (H)$ with $Ψ (z) = T$ .

A3A4

2.1

The two implications show that normality is equivalent to being the image of the coordinate function under a unital star representation of some $C (K)$ with nonempty compact $K \subseteq C$ ; the canonical instance is $K = σ (T)$ with the continuous functional calculus, and no other compact set is needed for the equivalence.

step 1.1step 1.2A5∎

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Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.70, printed pp.268–273 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, Example 4.9, pp.13–15 (standard reference, not scraped)