Contents

# Contents

## Definition

### On Hilbert spaces

A positive operator is a linear operator $A$ on a Hilbert space $(H, \langle -,-\rangle)$ such that the quadratic form

$v \mapsto \langle v, A v\rangle$

for $v \in H$ is positive semi-definite.

###### Proposition

Every positive operator $A$ on a Hilbert space is self-adjoint.

###### Proof

Let $B = \frac{1}{2}(A + A^\dagger)$ and $C = \frac{1}{2}i(A^\dagger - A)$. Then $B$ and $C$ are self-adjoint, and $A = B + iC$. Now, $\langle v, A v \rangle = \langle v, B v \rangle + i \langle v, C v \rangle$ is real for any $v$, so $\langle v, C v \rangle = 0$ for all $v$. Hence $C = 0$ and $A = B$.

### In $C^\ast$-algebras

More generally:

###### Definition

An element $A$ of an (abstract) C*-algebra is called positive if it is self-adjoint and its spectrum is contained in $[0, \infinity)$.

Here, ‘positive’ means positive semidefinite; see at inner product for the family of variations of this notion. (The relevant inner product here is that associated with the quadratic form above: $v, w \mapsto \langle v, A w\rangle$.)

### In dagger-categories

Yet more generally, in a dagger category a morphism $f$ is positive if it is of the form $f = g^\dagger \circ g$ for some morphism $g$.

## References

Discussion in dagger-compact categories with an eye twoards completely positive maps on spaces of density matrices:

Last revised on September 22, 2023 at 14:17:55. See the history of this page for a list of all contributions to it.