Positive operator (Hilbert space)

In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator [math]\displaystyle{ A }[/math] acting on an inner product space is called positive-semidefinite (or non-negative) if, for every [math]\displaystyle{ x \in \mathop{\text{Dom}}(A) }[/math], [math]\displaystyle{ \langle Ax, x\rangle \in \mathbb{R} }[/math] and [math]\displaystyle{ \langle Ax, x\rangle \geq 0 }[/math], where [math]\displaystyle{ \mathop{\text{Dom}}(A) }[/math] is the domain of [math]\displaystyle{ A }[/math]. Positive-semidefinite operators are denoted as [math]\displaystyle{ A\ge 0 }[/math]. The operator is said to be positive-definite, and written [math]\displaystyle{ A\gt 0 }[/math], if [math]\displaystyle{ \langle Ax,x\rangle\gt 0, }[/math] for all [math]\displaystyle{ x\in\mathop{\mathrm{Dom}}(A) \setminus \{0\} }[/math]. Many authors define a positive operator [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ \langle \cdot, \cdot \rangle }[/math] to be anti-linear on the first argument and linear on the second and suppose that [math]\displaystyle{ A }[/math] is positive and symmetric, the latter meaning that [math]\displaystyle{ \langle Ax,y \rangle= \langle x,Ay \rangle }[/math]. Then the non negativity of

[math]\displaystyle{ \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]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ \mu }[/math] shows that

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

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

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

For [math]\displaystyle{ x,y \in \mathop{\text{Dom}}A, }[/math] the polarization identity

[math]\displaystyle{ \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]\displaystyle{ \langle Ax,x\rangle = \langle x,Ax\rangle, }[/math] for positive operators, show that [math]\displaystyle{ \langle Ax,y\rangle = \langle x,Ay\rangle, }[/math] so [math]\displaystyle{ A }[/math] is symmetric.

In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space [math]\displaystyle{ H_\mathbb{R} }[/math] may not be symmetric. As a counterexample, define [math]\displaystyle{ A : \mathbb{R}^2 \to \mathbb{R}^2 }[/math] to be an operator of rotation by an acute angle [math]\displaystyle{ \varphi \in ( -\pi/2,\pi/2). }[/math] Then [math]\displaystyle{ \langle Ax,x \rangle = \|Ax\|\|x\|\cos\varphi \gt 0, }[/math] but [math]\displaystyle{ A^* = A^{-1} \neq A, }[/math] so [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ A }[/math] implies that [math]\displaystyle{ \mathop{\text{Dom}}A \subseteq \mathop{\text{Dom}}A^* }[/math] and [math]\displaystyle{ A = A^*|_{\mathop{\text{Dom}}(A)}. }[/math] For [math]\displaystyle{ A }[/math] to be self-adjoint, it is necessary that [math]\displaystyle{ \mathop{\text{Dom}}A = \mathop{\text{Dom}}A^*. }[/math] In our case, the equality of domains holds because [math]\displaystyle{ H_\mathbb{C} = \mathop{\text{Dom}}A \subseteq \mathop{\text{Dom}}A^*, }[/math] so [math]\displaystyle{ A }[/math] is indeed self-adjoint. The fact that [math]\displaystyle{ A }[/math] is bounded now follows from the Hellinger–Toeplitz theorem.

This property does not hold on [math]\displaystyle{ 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]\displaystyle{ B \geq A }[/math] if the following hold:

1. [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are self-adjoint
2. [math]\displaystyle{ 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.

Application to physics: quantum states

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

References

• Conway, John B. (1990), Functional Analysis: An Introduction, Springer Verlag, ISBN 0-387-97245-5 

• {{citation | last=Roman | first=Stephen