Positive operator

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

Many authors define a positive operator A {\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.

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 A {\displaystyle A} is positive and symmetric, the latter meaning that ⟨ A x , y ⟩ = ⟨ x , A y ⟩ {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle } . Then the non negativity of

⟨ A ( λ x + μ y ) , λ x + μ y ⟩ = | λ | 2 ⟨ A x , x ⟩ + λ ∗ μ ⟨ A x , y ⟩ + λ μ ∗ ⟨ A y , x ⟩ + | μ | 2 ⟨ A y , y ⟩ = | λ | 2 ⟨ A x , x ⟩ + λ ∗ μ ⟨ A x , y ⟩ + λ μ ∗ ( ⟨ A x , y ⟩ ) ∗ + | μ | 2 ⟨ A y , y ⟩ {\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 λ {\displaystyle \lambda } and μ {\displaystyle \mu } shows that

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

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

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

⟨ A x , y ⟩ = 1 4 ( ⟨ A ( x + y ) , x + y ⟩ − ⟨ A ( x − y ) , x − y ⟩ − i ⟨ A ( x + i y ) , x + i y ⟩ + i ⟨ A ( x − i y ) , x − i y ⟩ ) {\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 ⟨ A x , x ⟩ = ⟨ x , A x ⟩ , {\displaystyle \langle Ax,x\rangle =\langle x,Ax\rangle ,} for positive operators, show that ⟨ A x , y ⟩ = ⟨ x , A y ⟩ , {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,} so A {\displaystyle A} is symmetric.

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

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

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

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

1. A {\displaystyle A} and B {\displaystyle B} are self-adjoint
2. B − A ≥ 0 {\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]

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

• Conway, John B. (1990), Functional Analysis: An Introduction, Springer Verlag, ISBN 0-387-97245-5
• Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5