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, and , where 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 for all. Many authors define a positive operator A 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 A is positive and symmetric, the latter meaning that. Then the non negativity of for all complex \lambda and \mu shows that It follows that If A is defined everywhere, and then Ax = 0.

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 A 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 A 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 A implies that and For A to be self-adjoint, it is necessary that In our case, the equality of domains holds because 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

Partial order of self-adjoint operators

A natural partial ordering of self-adjoint operators arises from the definition of positive operators. Define B \geq A if the following hold: 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 and a set \cal S of positive trace-class operators \rho on for which The set \cal S is the set of states. Every is called a state or a density operator. For where 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 some sources define pure states to be unit elements from States that are not pure are called mixed.