Positive operator
Template:Short description In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator <math>A</math> acting on an inner product space is called positive-semidefinite (or non-negative) if, for every <math>x \in \operatorname{Dom}(A)</math>, <math>\langle Ax, x\rangle \in \mathbb{R}</math> and <math>\langle Ax, x\rangle \geq 0</math>, where <math>\operatorname{Dom}(A)</math> is the domain of <math>A</math>. Positive-semidefinite operators are denoted as <math>A\ge 0</math>. The operator is said to be positive-definite, and written <math>A>0</math>, if <math>\langle Ax,x\rangle>0,</math> for all <math>x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\}</math>.<ref>Template:Harvnb</ref>
Many authors define a positive operator <math>A </math> to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not imply self adjointness.
In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.
Cauchy–Schwarz inequalityEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Take the inner product <math>\langle \cdot, \cdot \rangle</math> to be anti-linear on the first argument and linear on the second and suppose that <math>A </math> is positive and symmetric, the latter meaning that <math> \langle Ax,y \rangle= \langle x,Ay \rangle </math>. Then the non negativity of
- <math>
\begin{align}
\langle A(\lambda x+\mu y),\lambda x+\mu y \rangle
=|\lambda|^2 \langle Ax,x \rangle + \lambda^* \mu \langle Ax,y \rangle+ \lambda \mu^* \langle Ay,x \rangle + |\mu|^2 \langle Ay,y \rangle \\[1mm] = |\lambda|^2 \langle Ax,x \rangle + \lambda^* \mu \langle Ax,y \rangle+ \lambda \mu^* (\langle Ax,y \rangle)^* + |\mu|^2 \langle Ay,y \rangle \end{align} </math> for all complex <math>\lambda </math> and <math> \mu </math> shows that
- <math>\left|\langle Ax,y\rangle \right|^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle.</math>
It follows that <math>\mathop{\text{Im}}A \perp \mathop{\text{Ker}}A.</math> If <math>A</math> is defined everywhere, and <math>\langle Ax,x\rangle = 0,</math> then <math>Ax = 0.</math>
On a complex Hilbert space, if an operator is non-negative then it is symmetricEdit
For <math>x,y \in \operatorname{Dom}A,</math> the polarization identity
- <math>
\begin{align} \langle Ax,y\rangle = \frac{1}{4}({} & \langle A(x+y),x+y\rangle - \langle A(x-y),x-y\rangle \\[1mm] & {} - i\langle A(x+iy),x+iy\rangle + i\langle A(x-iy),x-iy\rangle) \end{align} </math> and the fact that <math>\langle Ax,x\rangle = \langle x,Ax\rangle,</math> for positive operators, show that <math>\langle Ax,y\rangle = \langle x,Ay\rangle,</math> so <math>A</math> is symmetric.
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space <math>H_\mathbb{R}</math> may not be symmetric. As a counterexample, define <math>A : \mathbb{R}^2 \to \mathbb{R}^2</math> to be an operator of rotation by an acute angle <math>\varphi \in ( -\pi/2,\pi/2).</math> Then <math>\langle Ax,x \rangle = \|Ax\|\|x\|\cos\varphi > 0, </math> but <math>A^* = A^{-1} \neq A,</math> so <math>A</math> is not symmetric.
If an operator is non-negative and defined on the whole complex Hilbert space, then it is self-adjoint and boundedEdit
The symmetry of <math>A</math> implies that <math>\operatorname{Dom}A \subseteq \operatorname{Dom}A^*</math> and <math>A = A^*|_{\operatorname{Dom}(A)}.</math> For <math>A</math> to be self-adjoint, it is necessary that <math>\operatorname{Dom}A = \operatorname{Dom}A^*.</math> In our case, the equality of domains holds because <math>H_\mathbb{C} = \operatorname{Dom}A \subseteq \operatorname{Dom}A^*,</math> so <math>A</math> is indeed self-adjoint. The fact that <math>A</math> is bounded now follows from the Hellinger–Toeplitz theorem.
This property does not hold on <math>H_\mathbb{R}.</math>
Partial order of self-adjoint operatorsEdit
A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define <math>B \geq A</math> if the following hold:
- <math>A</math> and <math>B</math> are self-adjoint
- <math>B - A \geq 0</math>
It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.<ref>Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.</ref>
Application to physics: quantum statesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The definition of a quantum system includes a complex separable Hilbert space <math>H_\mathbb{C}</math> and a set <math>\cal S</math> of positive trace-class operators <math>\rho</math> on <math>H_\mathbb{C}</math> for which <math>\mathop{\text{Trace}}\rho = 1.</math> The set <math>\cal S</math> is the set of states. Every <math>\rho \in {\cal S}</math> is called a state or a density operator. For <math>\psi \in H_\mathbb{C},</math> where <math>\|\psi\| = 1,</math> the operator <math>P_\psi</math> of projection onto the span of <math>\psi</math> is called a pure state. (Since each pure state is identifiable with a unit vector <math>\psi \in H_\mathbb{C},</math> some sources define pure states to be unit elements from <math>H_\mathbb{C}).</math> States that are not pure are called mixed.