A positive operator is a linear operator on a Hilbert space such that the quadratic form
for is positive.
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: .)
Last revised on December 11, 2017 at 16:49:00. See the history of this page for a list of all contributions to it.