Spectral theorem for unbounded self-adjoint operators

Statement

Let $A$ be a self-adjoint operator on a complex Hilbert space $H$ . There is a unique projection-valued measure $E_A$ on the Borel subsets of $\mathbb R$ such that

A=\int_{\mathbb R}\lambda\,dE_A(\lambda)

as an unbounded spectral integral. Precisely,

\mathcal D(A)=\left\{\xi\in H: \int_{\mathbb R}\lambda^2\,d\langle E_A(\lambda)\xi,\xi\rangle<\infty\right\},

and the integral defines $A\xi$ on this domain. The measure is supported on the real spectrum of $A$ , so the spectral resolution determines both the operator and its domain.

Borel functional calculus

For a Borel function $f:\mathbb R\to\mathbb C$ , the theorem defines

f(A)=\int_{\mathbb R}f(\lambda)\,dE_A(\lambda)

on the vectors $\xi$ satisfying $\int |f(\lambda)|^2\,d\langle E_A(\lambda)\xi,\xi\rangle<\infty$ . Bounded $f$ give bounded operators on all of $H$ ; real-valued $f$ give self-adjoint operators. Indicator functions yield spectral projections, and this construction extends the continuous calculus to the Borel functional calculus.

Resolvents and reconstruction

For $z\in\mathbb C\setminus\mathbb R$ ,

(A-z)^{-1}=\int_{\mathbb R}(\lambda-z)^{-1}\,dE_A(\lambda).

Conversely, the spectral projections can be recovered from boundary behavior of the resolvent. The support of $E_A$ is $\sigma(A)$ , and a Borel set disjoint from the spectrum has zero spectral projection. Conversely, if an open set $U$ satisfies $E_A(U)=0$ , then $U$ is disjoint from the spectrum. These relations connect the measure-theoretic and resolvent forms of spectral theory Schmüdgen, Chapter 5.

Self-adjointness convention

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VIII on the spectral theorem and functional calculus for self-adjoint operators.
Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012. DOI record. Relevant: Chapters 4–5 on spectral measures, spectral integrals, and spectral decompositions.