Spectral measure of a normal operator

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.