Editing Positive operator

{{Short description|In mathematics, a linear operator acting on inner product space}}
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 a function|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>{{harvnb|Roman|2008|loc=p. 250 §10}}</ref>

Many authors define a '''positive operator''' <math>A </math> to be  a [[self-adjoint operator|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 state]]s, via the [[density matrix]] formalism.

== Cauchy–Schwarz inequality ==
{{Main|Cauchy–Schwarz inequality}}
Take  the inner product <math>\langle \cdot, \cdot \rangle</math> to be [[Antilinear map|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    symmetric ==
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 [[bounded operator|bounded]] ==
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 [[domain of a function|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 operators  ==
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 states ==
{{Main|Quantum state|Density operator}}
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]] [[density operator|operators]] <math>\rho</math> on <math>H_\mathbb{C}</math> for which <math>\mathop{\text{Trace}}\rho = 1.</math> The [[Set (mathematics)|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 [[Linear span|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 state (physics)|mixed]]''.

== References ==
{{Reflist}}
* {{Citation | last1=Conway | first1=John B.| title=Functional Analysis: An Introduction | publisher=[[Springer Verlag]] | isbn=0-387-97245-5 | year=1990}}
*{{citation | last=Roman | first=Stephen
 | title=Advanced Linear Algebra | edition=Third | series=[[Graduate Texts in Mathematics]] | publisher  = Springer | date=2008| pages= | isbn=978-0-387-72828-5 |author-link=Steven Roman}}

[[Category:Operator theory]]