Spectral integral - Knowlpedia

Core idea

Let $E:\Sigma\to\mathcal B(\mathcal H)$ be a projection-valued measure on $X$ , and let $f:X\to\mathbb C$ be measurable. The spectral integral

\int_X f\,dE

is first defined for simple $f=\sum_k a_k1_{A_k}$ by $\sum_k a_kE(A_k)$ . For general $f$ , it is the operator obtained by measurable approximation, with domain

D_f=\left\{\xi\in\mathcal H:\int_X|f|^2\,d\langle E(\,\cdot\,)\xi,\xi\rangle<\infty\right\}.

Here the displayed scalar integral is a Lebesgue integral, and this domain is part of the construction. If $f$ is essentially bounded relative to $E$ , the spectral integral is a bounded operator defined on all of $\mathcal H$ .

Algebraic and analytic properties

For bounded measurable functions, spectral integration is a unital $*$ -homomorphism into $\mathcal B(\mathcal H)$ : sums, products, complex conjugation, and uniform limits pass to the corresponding operator operations. Moreover,

\left\|\int_X f\,dE\right\|=\operatorname*{ess\,sup}_{E}|f|.

For unbounded $f$ , the spectral integral is a closed densely defined normal operator, and domain conditions are required before manipulating sums or products.

Functional calculus

If $A$ is self-adjoint with spectral measure $E_A$ , then its measurable functional calculus is

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

In particular, $A=\int_{\mathbb R}\lambda\,dE_A(\lambda)$ , where the unbounded coordinate function determines the domain. Characteristic functions recover spectral projections: $1_B(A)=E_A(B)$ .

Conventions and scope

The scalar measure $\langle E(\,\cdot\,)\xi,\xi\rangle$ depends on $\xi$ , so the displayed domain condition is stronger than ordinary scalar integrability against one fixed measure. “Spectral integral” here means integration against a projection-valued measure, not a general integral of Banach-space-valued functions. Equal functions modulo $E$ -null sets yield the same operator.

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. DOI record. Relevant: Chapter VII on spectral measures and functional calculus.
Nelson Dunford and Jacob T. Schwartz, Linear Operators, Part II: Spectral Theory, Wiley-Interscience, 1963. Publisher record. Relevant: Chapter X on spectral operators and operator-valued integration.