---
tag: K5VZ67T9J8
kind: concept
name: Spectral theory
msc: [47A10, 47B15, 47A60]
specializes: [YG9NR68ZH3]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_theory
same_as:
  wikidata: Q2409122
  wikipedia: Spectral theory
---

## Summary

Spectral theory studies linear operators through their spectra, the sets of numbers $\lambda$ for which $T - \lambda I$ fails to be invertible.
It extends the theory of eigenvalues and diagonalisation of matrices to operators on infinite-dimensional spaces.
Its central results, the spectral theorems, describe self-adjoint and normal operators as multiplication operators, and they are the mathematical basis of quantum mechanics.

## Explanations

### L1 human

A symmetric matrix can be described completely by its eigenvalues and eigenvectors: in the right coordinates, it just stretches each axis by a number.
Spectral theory asks how much of this survives for operators on spaces of functions or sequences.
For example, the operator that multiplies a square-integrable function $f(x)$ on $[0, 1]$ by $x$ has no eigenvectors at all, yet its spectrum is the whole interval $[0, 1]$.
Spectral theory gives the language to say in what sense this operator still "stretches by $x$ at the point $x$".