# 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

 $\langle Tv,v\rangle=\langle A^{*}Av,v\rangle=\langle Av,Av\rangle=\|Av\|^{2}\geq 0$

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

$T$ 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 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.

 Title positive element Canonical name PositiveElement Date of creation 2013-03-22 17:30:31 Last modified on 2013-03-22 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