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Let  be a complex Hilbert space. Let
 be a bounded operator in  .
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 simple to prove. Indeed,
Theorem -  is positive if and only if
The above notion can be generalized to elements in an arbitrary  -algebra.
In what follows
 denotes a  -algebra.
Definition - An element
 is said to be positive (and denoted  ) if
for some element
 .
 Positive elements are clearly self-adjoint.
It can be shown that the positive elements of
 are precisely the normal elements of
 with a positive spectrum. We state it here as a theorem:
Theorem - Let
 and  denote its spectrum. Then  is positive if and only if  is normal and
 .
Positive elements admit a unique positive square root.
Theorem - Let  be a positive element in
 . There is a unique
 such that
 is positive .
The converse is also true (with even weaker assumptions): If  admits a self-adjoint square root then  is positive, since
Another interesting fact about positive elements is that they form a proper convex cone 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  .
Theorem - The set of positive elements in
 is norm closed.
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