Appendix A: Functional calculus details

A. Functional Calculus Details

A.1 Borel functional calculus.

For $K$ selfadjoint with $E_K$, the map $\phi \mapsto \phi(K) = \int \phi\, dE_K$ is a $*$-homomorphism from bounded Borel functions on $\mathop{\mathrm{\sigma}}(K)$ into $B(\mathsf H)$, with $\|\phi(K)\| = \|\phi\|_{L^\infty(E_K)}$. Unbounded Borel $\phi$ gives densely defined $\phi(K)$, selfadjoint when $\phi$ is real.

A.2 Stone’s theorem.

Every strongly continuous one-parameter unitary group $(U_t)$ on $\mathsf H$ has a unique selfadjoint generator $\widetilde C$ with $U_t = e^{-it\widetilde C}$. Boundedness of $\widetilde C$ is equivalent to operator-norm continuity of $(U_t)$.