Borel functional calculus for self-adjoint and normal operators

Definition

Let $T$ be a self-adjoint operator, possibly unbounded, or a bounded normal operator on a Hilbert space, with spectral measure $E_T$ . In the unbounded self-adjoint case, this measure is supplied by the spectral theorem. For a complex Borel function $f$ on $\sigma(T)$ , its Borel functional calculus is

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

If $f$ is unbounded, this spectral integral has domain

\left\{\xi:\int_{\sigma(T)}|f(z)|^2\,d\langle E_T(z)\xi,\xi\rangle<\infty\right\}.

For bounded $f$ , it is a bounded operator on the whole Hilbert space. Functions equal outside an $E_T$ -null set determine the same operator.

Algebraic and convergence properties

On bounded Borel functions, $f\mapsto f(T)$ is a unital $*$ -homomorphism, modulo $E_T$ -null functions, and

\lVert f(T)\rVert=\operatorname*{ess\,sup}_{E_T}|f|.

Bounded pointwise convergence passes to strong-operator convergence by dominated convergence for the scalar measures $\langle E_T(\,\cdot\,)\xi,\xi\rangle$ . For unbounded functions, sums and products require domain control; identities from the bounded calculus cannot be transferred without taking the appropriate closures.

Relation to continuous functional calculus

For bounded normal $T$ , the continuous calculus sends $C(\sigma(T))$ into the norm-closed $C^*$ -algebra generated by $T$ and the identity. The bounded Borel calculus is generally larger: its values lie in the von Neumann algebra generated by the spectral projections. In particular, characteristic functions of Borel sets produce projections that need not belong to the norm-closed algebra. This is the essential strengthening beyond continuous functional calculus.

Self-adjoint and normal cases

For self-adjoint $T$ , the spectral measure is supported on $\mathbb R$ , and real-valued $f$ produce self-adjoint operators. For normal $T$ , the measure lives on $\mathbb C$ , and complex conjugation corresponds to adjunction. Taking $f(z)=z$ recovers $T$ , including its domain when $T$ is unbounded; taking $f=1_B$ gives the spectral projection for $B$ .

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised ed., Academic Press, 1980. DOI record. Relevant: Chapter VII on spectral measures and the measurable calculus.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III on spectral theory inside von Neumann algebras.