nLab
positive operator

Contents

Definition

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

vv,Av v \mapsto \langle v, A v\rangle

for vHv \in H is positive.

Proposition

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

Proof

Let B=12(A+A )B = \frac{1}{2}(A + A^\dagger) and C=12i(A A)C = \frac{1}{2}i(A^\dagger - A). Then BB and CC are self-adjoint, and A=B+iCA = B + iC. Now, v,Av=v,Bv+iv,Cv\langle v, A v \rangle = \langle v, B v \rangle + i \langle v, C v \rangle is real for any vv, so v,Cv=0\langle v, C v \rangle = 0 for all vv. Hence C=0C = 0 and A=BA = B.

More generally:

Definition

An element AA of an (abstract) C*-algebra is called positive if it is self-adjoint and its spectrum is contained in [0,)[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,wv,Awv, w \mapsto \langle v, A w\rangle.)

Last revised on December 11, 2017 at 11:49:00. See the history of this page for a list of all contributions to it.