---
tag: 77020CY5ZW
kind: concept
name: Positive operator
msc: [47B65, 47B15]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_operator
same_as:
  wikidata: Q2627013
  wikipedia: Positive operator
---

## Summary

A positive operator is a bounded self-adjoint operator $A$ on a Hilbert space with $\ip{Ax}{x} \ge 0$ for every vector $x$.
It is the operator version of a [[positive semidefinite matrix|01MMVW8CKC]] and of a nonnegative real number.
Positive operators define an order $A \le B$ on self-adjoint operators, they have [[square roots|3JY5HNFRMF]], and every operator of the form $T^*T$ is positive.

## Explanations

### L1 human

A positive operator never turns a vector to point against itself: the component of $Ax$ along $x$ is never negative.
On $\C^2$ the diagonal matrix with entries $1$ and $2$ is positive, since $\ip{Ax}{x} = \abs{x_1}^2 + 2\abs{x_2}^2 \ge 0$.
The diagonal matrix with entries $1$ and $-1$ is not, since it sends $(0, 1)$ to $(0, -1)$.
Positive operators play the role of the nonnegative numbers among operators.