nLab positive operator

Definition

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.

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 $A = B$.

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: $v, w \mapsto \langle v, A w\rangle$.)

Related concepts

• self-adjoint operator

• state on an operator algebra

• density matrix

• quantum effect

• quantum operation