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Positive operator

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In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every , and , where is the domain of . Positive-semidefinite operators are denoted as . The operator is said to be positive-definite, and written , if for all .[1]

Many authors define a positive operator 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

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Take the inner product to be anti-linear on the first argument and linear on the second and suppose that is positive and symmetric, the latter meaning that . Then the non negativity of

for all complex and shows that

It follows that If is defined everywhere, and then

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

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For the polarization identity

and the fact that for positive operators, show that so is symmetric.

In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space may not be symmetric. As a counterexample, define to be an operator of rotation by an acute angle Then but so is not symmetric.

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

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The symmetry of implies that and For to be self-adjoint, it is necessary that In our case, the equality of domains holds because so is indeed self-adjoint. The fact that is bounded now follows from the Hellinger–Toeplitz theorem.

This property does not hold on

Partial order of self-adjoint operators

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

  1. and are self-adjoint

It can be seen that a similar result as the Monotone convergence theorem holds for monotone increasing, bounded, self-adjoint operators on Hilbert spaces.[2]

Application to physics: quantum states

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The definition of a quantum system includes a complex separable Hilbert space and a set of positive trace-class operators on for which The set is the set of states. Every is called a state or a density operator. For where the operator of projection onto the span of is called a pure state. (Since each pure state is identifiable with a unit vector some sources define pure states to be unit elements from States that are not pure are called mixed.

References

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  1. ^ Roman 2008, p. 250 §10
  2. ^ Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.
  • Conway, John B. (1990), Functional Analysis: An Introduction, Springer Verlag, ISBN 0-387-97245-5
  • Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5