# positive element

## Primary tabs

Defines:
positive operator, positive cone, square root of positive element
Synonym:
positive
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### The converse

Hello,
in:
"The converse is also true, although it is not so simple to prove. Indeed,

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

do you assume that T is self-adjoint in the first place, so that <Tv,v> is real (and can therefore be compared to 0)? Or do you mean that [<Tv,v> is real and nonnegative] implies positivity for arbitrary T? (the latter sounds rather strong)

### Re: The converse

Hi,
I mean the latter: [<Tv,v> is real and nonnegative] implies positivity for arbitrary T.

Self-adjointness comes as a consequence. Notice that

<(T-T*)v,v> = <Tv,v> - <T*v,v> = <Tv,v> - <v,Tv> =

= <Tv,v> - conjugate{<Tv,v>} = <Tv,v> - <Tv,v> = 0

for all vectors v in H.

Since we are assuming that the Hilbert space is complex, we can conclude that T-T*= 0 , i.e. T is self-adjoint.

- Self-adjointness is really more than a consequence. In order to prove positivity we need to prove self-adjointness first (at least in the proofs I'm thinking about), so your question makes every sense.

To prove positivity you can either use the spectral theorem for normal operators (general version) or the theorem (stated in the entry) about the positive spectrum: T is positive iff [T is normal and has non-negative spectrum].

Hope this helps..

P.S. - I would dare to say that all of this is false for real Hilbert spaces.