Positive operator (Hilbert space) explained

A acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x \in Dom (A) , \langle Ax, x\rangle \in R and \langle Ax, x\rangle \geq 0 , where 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 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

See main article: Cauchy–Schwarz inequality. Take the inner product

\langle ⋅ , ⋅ \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( λ x + \mu y), λ x + \mu y \rangle = | λ | ^{2} \langle Ax,x \rangle + λ ^{*} \mu \langle Ax,y \rangle + λ \mu ^{*} \langle Ay,x \rangle + |\mu| ^{2} \langle Ay,y \rangle \\[ 1 mm] = | λ | ^{2} \langle Ax,x \rangle + λ ^{*} \mu \langle Ax,y \rangle + λ \mu ^{*} (\langle Ax,y \rangle) ^{*} + |\mu| ^{2} \langle Ay,y \rangle \end{align} for all complex λ and \mu shows that \left|\langle Ax,y\rangle \right| ^{2} \leq \langle Ax,x\rangle \langle Ay,y\rangle. It follows that Im A \perp 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 Dom A, the polarization identity\begin{align} \langle Ax,y\rangle = (1)/(4) ({} & \langle A(x + y),x + y\rangle - \langle A(x - y),x - y\rangle \\[ 1 mm] & {} - 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 _{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 \varphi \in ( - \pi/ 2 ,\pi/ 2 ). Then \langle Ax,x \rangle = \|Ax\|\|x\|\cos\varphi > 0 , but A ^{*} = A ^{-1} ≠ A, so A is not symmetric.

The symmetry of

A implies that Dom A \subseteq Dom A ^{*} and A = A ^{*|} _{Dom (A)} . For A to be self-adjoint, it is necessary that Dom A = Dom A ^{*.} In our case, the equality of domains holds because H _{C} = Dom A \subseteq 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 \geq A if the following hold:A and B are self-adjointB - 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.^[1]

Application to physics: quantum states

H _{C} and a set \cal S of positive trace-class operators \rho on H _{C} for which 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 _{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 _{C,} some sources define pure states to be unit elements from H _{C).} States that are not pure are called mixed.

Notes and References

1. Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.