Positive operator

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

Many authors define a positive operator $A$ 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

Template:Main Take the inner product $⟨\cdot ,\cdot ⟩$ to be anti-linear on the first argument and linear on the second and suppose that $A$ is positive and symmetric, the latter meaning that $⟨Ax,y⟩=⟨x,Ay⟩$. Then the non negativity of

$\begin{array}{c}⟨A(\lambda x+\mu y),\lambda x+\mu y⟩=|\lambda {|}^2⟨Ax,x⟩+{\lambda }^{*}\mu ⟨Ax,y⟩+\lambda {\mu }^{*}⟨Ay,x⟩+|\mu {|}^2⟨Ay,y⟩\\ =|\lambda {|}^2⟨Ax,x⟩+{\lambda }^{*}\mu ⟨Ax,y⟩+\lambda {\mu }^{*}(⟨Ax,y⟩{)}^{*}+|\mu {|}^2⟨Ay,y⟩\end{array}$

for all complex $\lambda$ and $\mu$ shows that

${|⟨Ax,y⟩|}^2\le ⟨Ax,x⟩⟨Ay,y⟩.$

It follows that $\text{Im}A\perp \text{Ker}A.$ If $A$ is defined everywhere, and $⟨Ax,x⟩=0,$ then $Ax=0.$

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

For $x,y\in \mathrm{Dom}A,$ the polarization identity

$\begin{array}{cc}⟨Ax,y⟩=\frac{1}{4}(& ⟨A(x+y),x+y⟩-⟨A(x-y),x-y⟩\\ & -i⟨A(x+iy),x+iy⟩+i⟨A(x-iy),x-iy⟩)\end{array}$

and the fact that $⟨Ax,x⟩=⟨x,Ax⟩,$ for positive operators, show that $⟨Ax,y⟩=⟨x,Ay⟩,$ so $A$ is symmetric.

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

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

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

This property does not hold on $H_{ℝ}.$

Partial order of self-adjoint operators

A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define $B\ge A$ if the following hold:

1. $A$ and $B$ are self-adjoint
2. $B-A\ge 0$

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

Application to physics: quantum states

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

References

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