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Positive operator - Wikipedia

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In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator {\displaystyle A} acting on an inner product space is called positive-semidefinite (or non-negative) if, for every {\displaystyle x\in \operatorname {Dom} (A)}, {\displaystyle \langle Ax,x\rangle \in \mathbb {R} } and {\displaystyle \langle Ax,x\rangle \geq 0}, where {\displaystyle \operatorname {Dom} (A)} is the domain of {\displaystyle A}. Positive-semidefinite operators are denoted as {\displaystyle A\geq 0}. The operator is said to be positive-definite, and written {\displaystyle A>0}, if {\displaystyle \langle Ax,x\rangle >0,} for all {\displaystyle x\in \mathop {\mathrm {Dom} } (A)\setminus \{0\}}.[1]

Many authors define a positive operator {\displaystyle A} 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 inequality

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Take the inner product {\displaystyle \langle \cdot ,\cdot \rangle } to be anti-linear on the first argument and linear on the second and suppose that {\displaystyle A} is positive and symmetric, the latter meaning that {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }. Then the non negativity of

{\displaystyle {\begin{aligned}\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{aligned}}}

for all complex {\displaystyle \lambda } and {\displaystyle \mu } shows that

{\displaystyle \left|\langle Ax,y\rangle \right|^{2}\leq \langle Ax,x\rangle \langle Ay,y\rangle .}

It follows that {\displaystyle \mathop {\text{Im}} A\perp \mathop {\text{Ker}} A.} If {\displaystyle A} is defined everywhere, and {\displaystyle \langle Ax,x\rangle =0,} then {\displaystyle Ax=0.}

On a complex Hilbert space, if an operator is non-negative then it is symmetric

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For {\displaystyle x,y\in \operatorname {Dom} A,} the polarization identity

{\displaystyle {\begin{aligned}\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{aligned}}}

and the fact that {\displaystyle \langle Ax,x\rangle =\langle x,Ax\rangle ,} for positive operators, show that {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,} so {\displaystyle A} is symmetric.

In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space {\displaystyle H_{\mathbb {R} }} may not be symmetric. As a counterexample, define {\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} to be an operator of rotation by an acute angle {\displaystyle \varphi \in (-\pi /2,\pi /2).} Then {\displaystyle \langle Ax,x\rangle =\|Ax\|\|x\|\cos \varphi >0,} but {\displaystyle A^{*}=A^{-1}\neq A,} so {\displaystyle A} is not symmetric.

If an operator is non-negative and defined on the whole Hilbert space, then it is self-adjoint and bounded

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The symmetry of {\displaystyle A} implies that {\displaystyle \operatorname {Dom} A\subseteq \operatorname {Dom} A^{*}} and {\displaystyle A=A^{*}|_{\operatorname {Dom} (A)}.} For {\displaystyle A} to be self-adjoint, it is necessary that {\displaystyle \operatorname {Dom} A=\operatorname {Dom} A^{*}.} In our case, the equality of domains holds because {\displaystyle H_{\mathbb {C} }=\operatorname {Dom} A\subseteq \operatorname {Dom} A^{*},} so {\displaystyle A} is indeed self-adjoint. The fact that {\displaystyle A} is bounded now follows from the Hellinger–Toeplitz theorem.

This property does not hold on {\displaystyle H_{\mathbb {R} }.}

Partial order of self-adjoint operators

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A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define {\displaystyle B\geq A} if the following hold:

  1. {\displaystyle A} and {\displaystyle B} are self-adjoint
  2. {\displaystyle B-A\geq 0}

It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.[2]

Application to physics: quantum states

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The definition of a quantum system includes a complex separable Hilbert space {\displaystyle H_{\mathbb {C} }} and a set {\displaystyle {\cal {S}}} of positive trace-class operators {\displaystyle \rho } on {\displaystyle H_{\mathbb {C} }} for which {\displaystyle \mathop {\text{Trace}} \rho =1.} The set {\displaystyle {\cal {S}}} is the set of states. Every {\displaystyle \rho \in {\cal {S}}} is called a state or a density operator. For {\displaystyle \psi \in H_{\mathbb {C} },} where {\displaystyle \|\psi \|=1,} the operator {\displaystyle P_{\psi }} of projection onto the span of {\displaystyle \psi } is called a pure state. (Since each pure state is identifiable with a unit vector {\displaystyle \psi \in H_{\mathbb {C} },} some sources define pure states to be unit elements from {\displaystyle H_{\mathbb {C} }).} States that are not pure are called mixed.

  1. ^ Roman 2008, p. 250 §10
  2. ^ Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.