---
tag: JB7S6WPZ77
kind: concept
name: Continuous functional calculus
msc: [46L05, 47A60, 46H30]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_functional_calculus
same_as:
  wikidata: Q1767119
  wikipedia: Continuous functional calculus
---

## Summary

Let $a$ be a normal element of a C*-algebra, and let $f$ be a continuous function on the [[spectrum|EH0CZ19RKP]] of $a$.
The continuous functional calculus gives a meaning to $f(a)$.
It extends the evaluation of polynomials at $a$ to all continuous functions, and it respects sums, products, adjoints and norms.
It is built from the [[commutative Gelfand–Naimark theorem|NT564Y315M]].
It is used to define square roots, absolute values, positive and negative parts, and exponentials of operators.

## Explanations

### L1 human

For a diagonal matrix, applying a function means applying it to each diagonal entry.
A matrix that can be diagonalized by a unitary change of basis can be handled the same way: diagonalize, apply the function to the eigenvalues, and change back.
The continuous functional calculus does this for normal elements of any C\*-algebra, such as normal operators on an infinite-dimensional Hilbert space.
For example, a positive operator has a unique positive square root, given by $f(t) = \sqrt{t}$.