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.
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
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.
The definition of a quantum system includes a complex separable Hilbert space and a set of positive trace-classoperators 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.
^Eidelman, Yuli, Vitali D. Milman, and Antonis Tsolomitis. 2004. Functional analysis: an introduction. Providence (R.I.): American mathematical Society.