Projection-valued measure

Let $(X,\Sigma)$ be a measurable space and $H$ a Hilbert space. Write $\mathcal P(H)$ for the set of orthogonal projections on $H$ . A projection-valued measure on $(X,\Sigma)$ is a map

E:\Sigma\longrightarrow\mathcal P(H)

such that $E(X)=I$ , $E(A\cap B)=E(A)E(B)$ , and, for every pairwise disjoint sequence $(A_n)$ and every $\xi\in H$ ,

E\left(\bigcup_{n=1}^{\infty}A_n\right)\xi =\sum_{n=1}^{\infty}E(A_n)\xi

with convergence in $H$ . Thus $E$ is countably additive in the strong operator topology. For $\xi,\eta\in H$ , the scalar function $A\mapsto\langle E(A)\xi,\eta\rangle$ is a countably additive complex measure.

Spectral integration

For a bounded measurable function $f$ , one first sets

\int_X\sum_j c_j1_{A_j}\,dE=\sum_j c_jE(A_j)

and then extends uniformly to define $\int_X f\,dE$ . This construction preserves products, adjoints, and constants, so it is a unital $*$ -homomorphism from bounded measurable functions into bounded operators. Indicator functions recover the projections: $\int_X1_A\,dE=E(A)$ .

Relation to spectral theorems

Every bounded normal operator $T$ has a unique projection-valued measure on its spectrum for which

T=\int_{\sigma(T)}z\,dE(z).

Conversely, spectral integration of the coordinate function produces a normal operator. For an unbounded self-adjoint operator, define $\mu_\xi(A)=\langle E(A)\xi,\xi\rangle$ ; the spectral integral is interpreted on the domain of vectors $\xi$ satisfying $\int_{\mathbb R}\lambda^2\,d\mu_\xi(\lambda)<\infty$ .

Examples and conventions

If $H=\bigoplus_nH_n$ is an orthogonal decomposition and $X=\mathbb N$ , then $E(A)$ projects onto $\bigoplus_{n\in A}H_n$ . This is the discrete model for spectral decomposition.

References

J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. Springer DOI record. Relevant: chapters on normal operators, spectral measures, and unbounded operators.