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A projection-valued measure on the Borel sets of $\mathbb{C}$ is a map $E$ into the orthogonal projections on a separable Hilbert space $\mathcal{H}$ such thatand for pairwise disjoint $B_j$,for every $v\in \mathcal{H}$, with convergence in norm.

The spectral theorem for normal operators on a separable Hilbert space states that a bounded normal operator $A$ has a unique projection-valued measure supported on $\mathrm{Sp}(A)$ for whichMore generally, the Borel functional calculus for a normal operator is

For $v,w\in \mathcal{H}$, the scalar spectral measures areIf $A$ is self-adjoint, its spectrum and hence the support of $E$ lie in $\mathbb{R}$. Moreover,so ${\mu }_v$ is a positive measure, andThe paper prints total mass $\Vert v\Vert$; with the standard definition it is $\Vert v{\Vert }^2$, so the unsquared norm is a typographical error.

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.