Definition - is said to be a positive operator if there exists a bounded operator such that
where denotes the adjoint of .
Every positive operator satisfies the very strong condition for every since
The converse is also true, although it is not so to prove. Indeed,
0.1 Generalization to -algebras
The above notion can be generalized to elements in an arbitrary -algebra (http://planetmath.org/CAlgebra).
In what follows denotes a -algebra.
Definition - An element is said to be positive (and denoted ) if
for some element .
0.2 Positive spectrum
It can be shown that the positive elements of are precisely the normal elements of with a positive spectrum. We it here as a theorem:
Theorem - Let and denote its spectrum. Then is positive if and only if is and .
0.3 Square roots
Positive elements admit a unique positive square root.
Theorem - Let be a positive element in . There is a unique such that
The converse is also true (with assumptions): If admits a square root then is positive, since
0.4 The positive cone
Another interesting fact about positive elements is that they form a proper convex cone (http://planetmath.org/Cone5) in , usually called the positive cone of . That is stated in following theorem:
Theorem - Let be positive elements in . Then
is also positive
is also positive for every
If both and are positive then .
0.5 Norm closure
Theorem - The set of positive elements in is norm closed.
|Date of creation||2013-03-22 17:30:31|
|Last modified on||2013-03-22 17:30:31|
|Last modified by||asteroid (17536)|
|Defines||square root of positive element|