Scalar spectral measure (${\mu }_{v,w}$)

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 $f$.