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 is the domain. The operator is said to be positive-definite.

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

to be anti-linear on the first. Then the non negativity of

shows that

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

the polarization identity

is symmetric.

to be an operator of rotation by an acute angle is not symmetric.

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

In our case, the equality of domains is bounded now follows from the Hellinger–Toeplitz theorem.

Partial order of self-adjoint operators

A natural partial ordering if the following hold:

1. 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.

Application to physics: quantum states

The definition of a quantum system includes a complex separable Hilbert space of positive trace-class operators The set is the set of states is called a state or a density operator of projection onto the span is called a pure state. (Since each pure state is identifiable with a unit vector States that are not pure are called mixed.

References