---
tag: BMKATJ802A
kind: concept
name: Spectrum of an operator
msc: [47A10]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
same_as:
  wikidata: Q1365748
  wikipedia: Spectrum (functional analysis)
---

## Summary

The spectrum of a bounded operator $T$ on a complex Banach space is the set of complex numbers $\lambda$ for which $T - \lambda I$ is not invertible.
It generalises the set of eigenvalues of a square matrix, the [[spectrum of a matrix|0XPX5NC67E]].
In infinite dimensions it can contain numbers that are not eigenvalues.
It is always a nonempty compact subset of the plane when the space is not zero.

## Explanations

### L1 human

For a matrix, $\lambda$ is an eigenvalue exactly when $A - \lambda I$ cannot be inverted.
The spectrum keeps this second description and uses it for operators in infinite dimensions.
There, $T - \lambda I$ can fail to be invertible even when it is injective, because it may miss some vectors.
For the right shift on square-summable sequences, there are no eigenvalues, yet the spectrum is the whole closed unit disc.