positive element

Let $H$ be a complex Hilbert space. Let $T:H\\longrightarrow H$ be a bounded operator in $H$ .

Definition - $T$ is said to be a positive operator if there exists a bounded operator $A:H\\longrightarrow H$ such that

T=A^{*}A

where $A^{*}$ denotes the adjoint of $A$ .

Every positive operator $T$ satisfies the very strong condition $\\langle Tv,v\\rangle\\geq 0$ for every $v\\in H$ since

\\langle Tv,v\\rangle=\\langle A^{*}Av,v\\rangle=\\langle Av,Av\\rangle=\\|Av\\|^{2}\\geq0

The converse is also true, although it is not so to prove. Indeed,

Theorem - $T$ is positive if and only if $\\langle Tv,v\\rangle\\geq 0\\;\\;\\;\\;\\forall_{v\\in H}$

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 $0\\leq x$ ) if

x=a^{*}a

for some element $a\\in\\mathcal{A}$ .

$Remark-$ Positive elements are clearly self-adjoint (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 $\\lambda\\geq 0$
•
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.