A positive operator is a linear operator on a Hilbert space such that the quadratic form
for is positive semi-definite.
Every positive operator on a Hilbert space is self-adjoint.
Let and . Then and are self-adjoint, and . Now, is real for any , so for all . Hence and .
More generally:
An element of an (abstract) C*-algebra is called positive if it is self-adjoint and its spectrum is contained in .
Here, ‘positive’ means positive semidefinite; see at inner product for the family of variations of this notion. (The relevant inner product here is that associated with the quadratic form above: .)
Yet more generally, in a dagger category a morphism is positive if it is of the form for some morphism .
(eg. Selinger 2005 Def. 4.1)
Discussion in dagger-compact categories with an eye twoards completely positive maps on spaces of density matrices:
Last revised on September 22, 2023 at 14:17:55. See the history of this page for a list of all contributions to it.