Spectral theorem for normal operators on a separable Hilbert space

A normal operator $A$ on a separable complex Hilbert space has a unique normalized projection-valued measure $E$ such thatFor a bounded normal operator, $E$ is supported on its compact spectrum and the integral defines an everywhere-defined bounded operator. For a closed densely defined normal operator, its operator domain isThe scalar measure in this expression is positive and has total mass $\Vert x{\Vert }^2$.

A spectral integral can exist without a single eigenvector. On $L^2(0,1)$, $(Af)(t)=tf(t)$ is bounded and self-adjoint, and $E(B)f=1_{B}f$. An eigenvector for $\lambda$ would have to be supported on $\{t=\lambda \}$, a set of Lebesgue measure zero, so it would vanish in the L2 space. The spectral measure therefore extends orthonormal diagonalization to continuous spectral parameters rather than claiming an eigenbasis always exists.

Every self-adjoint operator on a separable complex Hilbert space is diagonal modulo a compact operator. For a bounded operator the compact self-adjoint error can have arbitrarily small Hilbert-Schmidt norm. In a cyclic spectral representation, partition a bounded real interval into $N$ intervals of width at most $\delta$ and successively bisect. Orthogonal step-function differences give basis vectors supported in the parent intervals. A diagonal value chosen in each support interval gives total squared column error at most $3N{\delta }^2$. Since $N=O({\delta }^{-1})$, this tends to zero. Apply the construction to countably many cyclic summands with summable squared error bounds. For an unbounded operator first split into bounded spectral intervals, obtaining a bounded compact error on the same domain.

Bounded self-adjoint operators on separable infinite-dimensional complex Hilbert spaces are unitarily equivalent modulo compact operators exactly when they have the same essential spectrum in the Calkin algebra. The Weyl-von Neumann theorem diagonalizes each modulo a compact error. Their diagonal cluster sets are the essential spectra; cluster-set matching of bounded sequences pairs the two diagonal lists with differences tending to zero, giving a compact diagonal difference. The separability assumption is essential: projections with countable and uncountable range dimensions can have the same essential spectrum but cannot differ by a compact operator after unitary conjugation.

A projection-valued measure $E$ assigns an orthogonal projection $E(B)$ to each measurable set, with $E(\varnothing )=0$, $E(X)=I$, $E(B\cap C)=E(B)E(C)$, and countable additivity in the strong operator topology on pairwise disjoint sets.

The continuous functional calculus of a bounded normal operator $T$ is a representation of $C(K)$, with $K=\sigma (T)$. The Riesz-Markov-Kakutani representation theorem represents $f\mapsto ⟨f(T)v,v⟩$ by a positive regular measure ${\mu }_v$. The assignment $f\mapsto f(T)v$ is isometric for the $L^2({\mu }_v)$ norm, and extends to a unitary map onto the closed cyclic subspace it generates. Multiplication by $z$ and $\overline{z}$ represent $T$ and $T^{\ast }$. Orthogonal decomposition into cyclic reducing subspaces, allowing an uncountable family, constructs the Borel functional calculus for a normal operator without requiring separability.

For a unital star-representation $\pi :C(K)\to \mathcal{B}(H)$, the Riesz-Markov-Kakutani representation theorem gives regular measures ${\mu }_{x,y}$ representing $f\mapsto ⟨\pi (f)x,y⟩$. Define $P(E)$ by $⟨P(E)x,y⟩={\mu }_{x,y}(E)$. Positivity and ${\mu }_{x,x}(K)=\Vert x{\Vert }^2$ make these positive contractions. The identities ${\mu }_{\pi (f)x,y}=f{\mu }_{x,y}$ imply commutation with $\pi (f)$ and ${\mu }_{P(E)x,y}=1_E{\mu }_{x,y}$; hence $P(F)P(E)=P(F\cap E)$. Thus each $P(E)$ is an orthogonal projection. Scalar countable additivity and orthogonality give countable additivity in the strong operator topology, constructing the normalized regular projection-valued measure.

If $A$ is a commutative C-star algebra of bounded operators on a Hilbert space containing its identity, there is a unique regular projection-valued measure $E$ on the character space ${\Phi }_A$ such that $E({\Phi }_A)=I$ and $a={\int }_{{\Phi }_A}\hat{a}dE$ for every $a\in A$, where $\hat{a}$ is the Gelfand transform.

For a faithful representation of $C(K)$ by integration against a regular projection-valued measure $P$, every nonempty open subset $U$ satisfies $P(U)\ne 0$. Choose a nonzero continuous function supported inside $U$, using compact Hausdorff normality. If $P(U)=0$, its spectral integral vanishes, contradicting faithfulness. Regularity is understood through the scalar spectral measures.

A spectral projector is the value $E(B)$ of the projection-valued measure of a normal operator on a measurable part $B$ of its spectrum. In finite dimension, the spectral projector onto an eigenspace extracts the corresponding eigenvector component.

For a normal operator $A$ with projection-valued measure $E$ and vectors $f,g$, its scalar spectral measure of a normal operator isThis is generally a complex measure, while ${\mu }_f={\mu }_{f,f}$ is positive with total mass $\Vert f{\Vert }^2$. For every bounded Borel function $h$, the Borel functional calculus for a normal operator satisfiesFor an unbounded $h$, one must additionally have $f\in D(h(A))$, equivalently $\int |h{|}^{2}d{\mu }_f<\infty$. This domain condition makes the pairing and scalar integral well-defined; an arbitrary unbounded function cannot be inserted without checking it.

For a self-adjoint operator $A$, its projection-valued spectral measure of a normal operator assigns each Borel set $S\subseteq \mathbb{R}$ an orthogonal projection $E_A(S)$. For a normalized quantum state $\psi$, the probability of a measurement result in $S$ is $⟨\psi ,E_A(S)\psi ⟩$. In finite dimensions, $E_A(S)$ is the sum of the orthogonal projections onto eigenspaces whose eigenvalues lie in $S$. This generalizes an individual eigenvector probability to degenerate eigenvalues and continuous spectra. The functional-calculus notation $1_S(A)$ uses the indicator function of the set.

For a bounded normal operator $T$ with spectral measure of a normal operator $P$, any projection $P(E)$ commutes with $T$ and $T^{\ast }$ by the Borel functional calculus for a normal operator. Its range is a closed reducing subspace. If the spectrum has two points, choose disjoint nonempty relative open sets around them. Full support of a faithful spectral measure makes both projections nonzero and their product zero, so either range is nonzero and proper, in particular an invariant subspace.

For a projection-valued measure $E$ and vectors $v,w$, the scalar spectral measure is the complex measure ${\mu }_{v,w}(B)=⟨E(B)v,w⟩$. The positive measure ${\mu }_v={\mu }_{v,v}$ has total mass $\Vert v{\Vert }^2$.

A sequence of scalar spectral measures ${\mu }_{v,w}^{(n)}$ converges weakly to ${\mu }_{v,w}$ if its integrals against every bounded continuous function converge. A concrete sufficient setting is that all underlying normal-operator spectra lie in one fixed compact set $K\subset \mathbb{C}$ andThe Stone-Weierstrass theorem approximates any continuous function on $K$ uniformly by polynomials in $z,\overline{z}$. Also $\Vert {\mu }_{v,w}^{(n)}{\Vert }_{\mathrm{TV}}\le \Vert v\Vert \Vert w\Vert$: apply Cauchy-Schwarz to the disjoint spectral projections in any finite partition. These uniform total-variation bounds control the approximation errors, so the mixed polynomial tests imply all continuous tests.

On a noncompact real spectrum, a uniform second-moment bound gives uniform tightness for positive scalar spectral measures, but does not alone justify the polynomial approximation step. High-degree polynomial tails require additional uniform integrability, and identifying a limit from all moments needs an additional hypothesis ensuring that these determine the measure. Merely asserting tightness does not supply either missing condition.

For a self-adjoint operator $A$, fix the resolvent convention $R(z)=(A-zI{)}^{-1}$. For real $a<b$, Stone formula states the strong-operator limitEquivalently the integrand is ${\pi }^{-1}\mathrm{Im}R(t+iϵ)$. The endpoint halves are necessary when there are eigenvalues at the endpoints; if neither endpoint has a spectral atom, the limit is simply the interval spectral projection.^[ref]

To see the limit, use the spectral theorem to replace the resolvent difference by the scalar kernelIt lies in $[0,1]$. Its arctangent primitive tends to one inside $(a,b)$, zero outside $[a,b]$, and one half at each endpoint. For every vector $x$, dominated convergence applied to $\int |F_{ϵ}-F_0{|}^{2}d{\mu }_x$ proves convergence in norm after applying the operators to $x$. This explains both the strong rather than necessarily operator-norm limit and the half weights.