---
tag: 92433H25TC
kind: concept
name: Borel functional calculus
msc: [47A60, 47B15]
sources:
  - url: https://en.wikipedia.org/wiki/Borel_functional_calculus
same_as:
  wikidata: Q829645
  wikipedia: 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 $N$.

## Explanations

### L1 human

For a diagonal matrix, applying a function to the matrix means applying it to each diagonal entry.
The spectral theorem says that normal operators are diagonal in a generalized sense, with a measure in place of a list of eigenvalues.
The Borel functional calculus applies a function to an operator in the same way, entry by entry along the spectrum.
For the matrix $\operatorname{diag}(1, 2)$ and the indicator function $f$ of the set $\set{2}$, the result is $f(N) = \operatorname{diag}(0, 1)$, the projection onto the eigenspace of $2$.