---
tag: EH0CZ19RKP
kind: concept
name: Spectrum of an element
msc: [46H05, 47A10]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
---

## Summary

The spectrum of an element $a$ of a unital algebra is the set of scalars $\lambda$ for which $a - \lambda 1$ has no inverse.
For square matrices it is the set of eigenvalues, and for bounded operators it can be larger than the set of eigenvalues.
In a complex Banach algebra the spectrum is a nonempty compact set.
It is the starting point of spectral theory and of the functional calculus.

## Explanations

### L1 human

For a matrix $a$, the number $\lambda$ is an eigenvalue exactly when $a - \lambda$ cannot be inverted.
The spectrum keeps this second description and uses it in any algebra where one can multiply and invert.
For a continuous function $f$ on $[0,1]$, the function $f - \lambda$ has an inverse exactly when it never vanishes.
So the spectrum of $f$ is the set of values of $f$.