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

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

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

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

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

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

  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
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