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 $x \in \mathop(A)$, $\langle Ax, x\rangle \in \mathbb$ and $\langle Ax, x\rangle \geq 0$, where $\mathop(A)$ 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(A) \setminus \$.

In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.

Cauchy–Schwarz inequality

If $A \geq 0,$ then

:$\left, \langle Ax,y\rangle \^2 \leq \langle Ax,x\rangle \langle Ay,y\rangle.$

Indeed, let $\varepsilon > 0.$ Applying Cauchy–Schwarz inequality to the inner product

:$(x,y)_\varepsilon \stackrel\ \langle (A+\varepsilon\cdot \mathbf)x,y\rangle$

as $\varepsilon \downarrow 0$ proves the claim.

It follows that $\mathopA \perp \mathopA.$ If $A$ is defined everywhere, and $\langle Ax,x\rangle = 0,$ then $Ax = 0.$

On ''H'', if ''A'' ≥ 0 then ''A'' is symmetric

Without loss of generality, let the inner product $\langle \cdot, \cdot \rangle$ be anti-linear on the ''first'' argument and linear on the second. (If the reverse is true, then we work with $\langle x,y\rangle_\text \stackrel \ \langle y,x\rangle$ instead). For $x,y \in \mathopA,$ the polarization identity
:
\begin
\langle Ax,y\rangle = \frac( & \langle A(x+y),x+y\rangle - \langle A(x-y),x-y\rangle \\ mm& - i\langle A(x+iy),x+iy\rangle + i\langle A(x-iy),x-iy\rangle)
\end

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 $H_\mathbb$ may not be symmetric. As a counterexample, define $A : \mathbb^2 \to \mathbb^2$ 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 $A^* = A^ \neq A,$ so $A$ is not symmetric.

If ''A'' ≥ 0 and Dom ''A'' = ''H'', then ''A'' is self-adjoint and bounded

The symmetry of $A$ implies that $\mathopA \subseteq \mathopA^*$ and $A = A^*, _.$ For $A$ to be self-adjoint, it is necessary that $\mathopA = \mathopA^*.$ In our case, the equality of domains holds because $H_\mathbb = \mathopA \subseteq \mathopA^*,$ 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 $H_\mathbb.$

Order in self-adjoint operators on ''H''

A natural 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-adjoint
# $B - A \geq 0$

It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.

Application to physics: quantum states

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

References

* {{Citation , last1=Conway , first1=John , title=Functional Analysis: An Introduction , publisher=Springer Verlag , isbn=0-387-97245-5 , year=1990

Operator theory