positive element

Let H be a complex Hilbert spaceMathworldPlanetmath. Let T:HH be a bounded operatorMathworldPlanetmathPlanetmath in H.

Definition - T is said to be a positive operator if there exists a bounded operator A:HH such that


where A* denotes the adjoint of A.

Every positive operator T satisfies the very strong condition Tv,v0 for every vH since


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

Theorem - T is positive if and only if Tv,v0vH

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 𝒜 denotes a C*-algebra.

Definition - An element x𝒜 is said to be positive (and denoted 0x) if


for some element a𝒜.

Remark- Positive elementsMathworldPlanetmathPlanetmath are clearly self-adjoint (http://planetmath.org/InvolutaryRing).

0.2 Positive spectrum

It can be shown that the positive elements of 𝒜 are precisely the normal elements of 𝒜 with a positive spectrum. We it here as a theorem:

Theorem - Let x𝒜 and σ(x) denote its spectrum. Then x is positive if and only if x is and σ(x)0+.

0.3 Square roots

Positive elements admit a unique positive square root.

Theorem - Let x be a positive element in 𝒜. There is a unique b𝒜 such that

  • b is positive

  • x=b2.

The converse is also true (with assumptionsPlanetmathPlanetmath): If x admits a square root then x is positive, since


0.4 The positive cone

Another interesting fact about positive elements is that they form a proper convex cone (http://planetmath.org/Cone5) in 𝒜, usually called the positive conePlanetmathPlanetmath of 𝒜. That is stated in following theorem:

Theorem - Let a,b be positive elements in 𝒜. Then

  • a+b is also positive

  • λa is also positive for every λ0

  • If both a and -a are positive then a=0.

0.5 Norm closure

Theorem - The set of positive elements in 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