Positive operator (Hilbert space)

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 . In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism. Cauchy–Schwarz inequality If then Indeed, let Applying Cauchy–Schwarz inequality to the inner product as proves the claim. It follows that If is defined everywhere, and then Without loss of generality, let the inner product be anti-linear on the first argument and linear on the second. (If the reverse is true, then we work with instead). 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. 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 A natural ordering of self-adjoint operators arises from the definition of positive operators. Define if the following hold: 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. Quantum state and Density operator 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.