Positive operator
In mathematics as well as physics, a linear operator acting on an inner product space is called positive-semidefinite 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.
Many authors define a positive operator to be a self-adjoint 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, 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 identityand 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 complex Hilbert space, then it is self-adjoint and bounded">bounded operator">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:- and are self-adjoint