Borel functional calculus

Summary

The Borel functional calculus assigns an operator $f(N)$ to a normal or self-adjoint operator $N$ and a Borel function $f$ on its spectrum. It is defined by integrating $f$ against the projection-valued measure given by the spectral theorem. It extends the continuous functional calculus to discontinuous functions, such as indicator functions, which produce the spectral projections of $N$.