---
tag: YG9NR68ZH3
kind: concept
name: Operator theory
msc: [47A05, 47B01, 47B02]
sources:
  - url: https://en.wikipedia.org/wiki/Operator_theory
same_as:
  wikidata: Q1198874
  wikipedia: Operator theory
---

## Summary

Operator theory is the part of functional analysis that studies linear operators between normed spaces, above all bounded operators on Banach and Hilbert spaces.
It asks how far the theory of matrices, with eigenvalues, adjoints and diagonalisation, extends to infinite dimensions.
Its tools are used for differential and integral equations, quantum mechanics and signal processing.

## Explanations

### L1 human

A matrix turns a vector of numbers into another vector of numbers.
An operator does the same thing for objects with infinitely many coordinates, such as sequences or functions.
A good picture is the derivative, which turns a function into another function and respects sums and scalar multiples.
A simple example is the right shift on sequences, $(x_1, x_2, \ldots) \mapsto (0, x_1, x_2, \ldots)$.
It never loses information, yet it is not onto, something no square matrix can do.
Operator theory explains such new effects of infinite dimension.