Borel functional calculus for bounded normal operators

Clauses 1 to 4: the agreement with the continuous calculus and ${\mathbf{1}}_B(T)=E(B)$ are the definition of the Borel calculus and the identification ${\mathrm{\Phi }}_E({\mathbf{1}}_B)=E(B)$; linearity, unitality, multiplicativity and conjugation preservation, the quadratic identity, the norm formula and the strong-convergence property are the corresponding properties of ${\mathrm{\Phi }}_E$; normality follows because $f(T)f(T{)}^{\ast }={\mathrm{\Phi }}_E(f\bar{f})={\mathrm{\Phi }}_E(\bar{f}f)=f(T{)}^{\ast }f(T)$.

The measures ${\mu }_{Sx,y}$ and ${\mu }_{x,S^{\ast }y}$ coincide: for every continuous $f$ one has $\int fd{\mu }_{Sx,y}=⟨f(T)Sx,y⟩=⟨Sf(T)x,y⟩=⟨f(T)x,S^{\ast }y⟩=\int fd{\mu }_{x,S^{\ast }y}$, using the commutant clause of the continuous calculus and the adjoint identity; both are finite regular complex measures, so equality of their integrals against all continuous functions gives ${\mu }_{Sx,y}={\mu }_{x,S^{\ast }y}$.

The commutant clause: for every bounded Borel $h$ and all $x,y\in H$ one has $⟨h(T)Sx,y⟩=\int hd{\mu }_{Sx,y}=\int hd{\mu }_{x,S^{\ast }y}=⟨h(T)x,S^{\ast }y⟩=⟨Sh(T)x,y⟩$, so $Sh(T)=h(T)S$; in particular $S$ commutes with every spectral projection $E(B)={\mathbf{1}}_B(T)$ and with every Borel calculus operator.

All five clauses hold: the Borel calculus extends the continuous calculus, is a unital star-homomorphism with $E$-essential-supremum norm, is strongly continuous under uniformly bounded pointwise $E$-almost everywhere convergence, and every operator commuting with $T$ and $T^{\ast }$ commutes with all of it.