= Weak convergence of scalar spectral measures
A sequence of <scalar spectral measures> $\mu^{(n)}_{v,w}$ 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 $K\subset\mathbb C$ and
$$
\int z^p\overline z^{\,q}\,d\mu^{(n)}_{v,w}\longrightarrow\int z^p\overline z^{\,q}\,d\mu_{v,w}\qquad(p,q\geq0).
$$
The <Stone-Weierstrass theorem> approximates any continuous function on $K$ uniformly by polynomials in $z,\overline z$. Also $\|\mu^{(n)}_{v,w}\|_{\mathrm{TV}}\leq\|v\|\|w\|$: 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.
Back to article page
