Positive operator (Hilbert space)

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 $$x \in \mathop{\text{Dom}}(A)$$, $$\langle Ax, x\rangle \in \mathbb{R}$$ and $$\langle Ax, x\rangle \geq 0$$, where $$\mathop{\text{Dom}}(A)$$ 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 $$\langle Ax,x\rangle>0,$$ for all $$x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\}$$.

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
Take the inner product $$\langle \cdot, \cdot \rangle$$ 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   $$  \langle Ax,y \rangle= \langle x,Ay \rangle $$. Then the non negativity of

\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} $$ for all complex $$\lambda $$ and $$ \mu $$ shows that
 * $$\left|\langle Ax,y\rangle \right|^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle.$$

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

On a complex Hilbert space, if an operator is non-negative then it is   symmetric
For $$x,y \in \mathop{\text{Dom}}A,$$ the polarization identity



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

In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space $$H_\mathbb{R}$$ may not be symmetric. As a counterexample, define $$A : \mathbb{R}^2 \to \mathbb{R}^2$$ to be an operator of rotation by an acute angle $$\varphi \in ( -\pi/2,\pi/2).$$ Then $$\langle Ax,x \rangle = \|Ax\|\|x\|\cos\varphi > 0, $$ but $$A^* = A^{-1} \neq A,$$ 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 $$\mathop{\text{Dom}}A \subseteq \mathop{\text{Dom}}A^*$$ and $$A = A^*|_{\mathop{\text{Dom}}(A)}.$$ For $$A$$ to be self-adjoint, it is necessary that $$\mathop{\text{Dom}}A = \mathop{\text{Dom}}A^*.$$ In our case, the equality of domains holds because $$H_\mathbb{C} = \mathop{\text{Dom}}A \subseteq \mathop{\text{Dom}}A^*,$$ 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 $$H_\mathbb{R}.$$

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


 * 1) $$A$$ and $$B$$ are self-adjoint
 * 2) $$B - A \geq 0$$

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

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