Spectral measure of a normal operator

For vectors $f,g$, the scalar spectral measure is ${\mu }_{f,g}(S)=⟨E(S)f,g⟩$. Functional calculus satisfies $⟨h(A)f,g⟩=\int hd{\mu }_{f,g}$.

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 $\Vert v{\Vert }^2$.

Scalar spectral measures ${\mu }_{v,w}^{(n)}$ converge weakly to ${\mu }_{v,w}$ when $\int fd{\mu }_{v,w}^{(n)}\to \int fd{\mu }_{v,w}$ for every bounded continuous function $f$. Uniform second-moment bounds supply the tightness needed to pass from polynomial tests to all such $f$.

Stone's formula recovers spectral projections of a self-adjoint operator from the jump of its resolvent across the real axis, with half weight at interval endpoints.