positive element
Let $H$ be a complex Hilbert space^{}. Let $T:H\u27f6H$ be a bounded operator^{} in $H$.
Definition  $T$ is said to be a positive operator if there exists a bounded operator $A:H\u27f6H$ such that
$$T={A}^{*}A$$ 
where ${A}^{*}$ denotes the adjoint of $A$.
Every positive operator $T$ satisfies the very strong condition $\u27e8Tv,v\u27e9\ge 0$ for every $v\in H$ since
$$\u27e8Tv,v\u27e9=\u27e8{A}^{*}Av,v\u27e9=\u27e8Av,Av\u27e9={\parallel Av\parallel}^{2}\ge 0$$ 
The converse^{} is also true, although it is not so to prove. Indeed,
Theorem  $T$ is positive if and only if $\u27e8Tv,v\u27e9\ge 0\mathit{\hspace{1em}}{\forall}_{v\in H}$
0.1 Generalization to ${C}^{*}$algebras
The above notion can be generalized to elements in an arbitrary ${C}^{*}$algebra (http://planetmath.org/CAlgebra).
In what follows $\mathcal{A}$ denotes a ${C}^{*}$algebra.
Definition  An element $x\in \mathcal{A}$ is said to be positive (and denoted $0\le x$) if
$$x={a}^{*}a$$ 
for some element $a\in \mathcal{A}$.
$Remark$ Positive elements^{} are clearly selfadjoint (http://planetmath.org/InvolutaryRing).
0.2 Positive spectrum
It can be shown that the positive elements of $\mathcal{A}$ are precisely the normal elements of $\mathcal{A}$ with a positive spectrum. We it here as a theorem:
Theorem  Let $x\in \mathcal{A}$ and $\sigma (x)$ denote its spectrum. Then $x$ is positive if and only if $x$ is and $\sigma (x)\subset {\mathbb{R}}_{0}^{+}$.
0.3 Square roots
Positive elements admit a unique positive square root.
Theorem  Let $x$ be a positive element in $\mathcal{A}$. There is a unique $b\in \mathcal{A}$ such that

•
$b$ is positive

•
$x={b}^{2}$.
The converse is also true (with assumptions^{}): If $x$ admits a square root then $x$ is positive, since
$$x={b}^{2}=bb={b}^{*}b$$ 
0.4 The positive cone
Another interesting fact about positive elements is that they form a proper convex cone (http://planetmath.org/Cone5) in $\mathcal{A}$, usually called the positive cone^{} of $\mathcal{A}$. That is stated in following theorem:
Theorem  Let $a,b$ be positive elements in $\mathcal{A}$. Then

•
$a+b$ is also positive

•
$\lambda a$ is also positive for every $\lambda \ge 0$

•
If both $a$ and $a$ are positive then $a=0$.
0.5 Norm closure
Theorem  The set of positive elements in $\mathcal{A}$ is norm closed.
Title  positive element 
Canonical name  PositiveElement 
Date of creation  20130322 17:30:31 
Last modified on  20130322 17:30:31 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  8 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46L05 
Classification  msc 47L07 
Classification  msc 47A05 
Synonym  positive 
Defines  positive operator 
Defines  positive cone 
Defines  square root of positive element 