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{{Cleanup|reason=This page is about positive Hilbert space operators. The discussion on positive operators on ordered Banach spaces probably does not belong here.|date=October 2013}}


In [[mathematics]], especially [[functional analysis]], a [[self-adjoint]] (or [[hermitian]]) element <math>A</math> of a [[C*-algebra]] <math>\mathcal{A}</math> is called '''positive''' if its [[spectrum of an operator|spectrum]] <math>\sigma(A)</math> consists of non-negative real numbers. Moreover, an element <math>A</math> of a C*-algebra <math>\mathcal{A}</math> is positive if and only if there is some <math>B</math> in <math>\mathcal{A}</math> such that <math>A = B^* B</math>. A positive element is self-adjoint and thus [[normal element|normal]].


If <math>T</math> is a [[bounded operator|bounded]] [[linear operator]] on a [[Hilbert space]] <math>H</math>, then this notion coincides with the condition that <!-- <math>T</math> is self-adjoint and --> <!-- No need to demand this explicitly, "self-adjointness" follows automatically!--> <math>\langle Tx,x \rangle</math> is non-negative for every vector <math>x</math> in <math>H</math>. <!-- Is "boundedness" needed here? (Yes, boundedness is equivalent to continuity) Doesn't positivity imply continuity? (No, there are non-positive linear operators, see below). You mean, there are unbounded positive linear operators. Anyway, in the first place positivity is defined for operators that are bounded.--> Note that <math>\langle Tx,x \rangle</math> is real for every <math>x</math> in <math>H</math> if and only if <math>T</math>' is self-adjoint. <!--This equivalence is a standard result in the theory of operators on Hilbert spaces. Reference needed?--> Hence, a positive operator on a Hilbert space is always [[self-adjoint operator|self-adjoint]] (and a self-adjoint ''everywhere defined'' operator on a Hilbert space is always bounded because of the [[Hellinger-Toeplitz theorem]]).
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The set of positive elements of a C*-algebra forms a [[convex cone]].
 
== Positive and positive definite operators ==
A bounded linear operator <math>P</math> on an [[inner product space]] <math>V</math> is said to be ''positive'' (or ''positive semidefinite'') if <math>P = S^*S</math> for some bounded operator <math>S</math> on <math>V</math>, and is said to be ''positive definite'' if <math>S</math> is also [[non-singular operator|non-singular]]. <!-- link to a page on non-singular operators needed!-->
 
<!--Relevant theorems:-->
'''(I)''' The following conditions for a bounded operator <math>P</math> on <math>V</math> to be positive semidefinite are equivalent:
* <math>P = S^*S</math> for some bounded operator <math>S</math> on <math>V</math>,
* <math>P = T^2</math> for some self-adjoint operator <math>T</math> on <math>V</math>,
* <math>P</math> is self-adjoint and <math>\langle Pu,u \rangle \geq 0, \forall u \in V</math>.
 
'''(II)''' The following conditions for a bounded operator <math>P</math> on <math>V</math> to be positive definite are equivalent:
* <math>P = S^*S</math> for some non-singular bounded operator <math>S</math> on <math>V</math>,
* <math>P = T^2</math> for some non-singular self-adjoint operator <math>T</math> on <math>V</math>,
* <math>P</math> is self adjoint and <math>\langle Pu,u \rangle > 0, \forall u \neq 0</math> in <math>V</math>.
 
'''(III)''' A complex matrix <math>A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}</math> represents a positive (semi)definite operator if and only if <math>A</math> is [[hermitian matrix|hermitian]] (or self-adjoint) <!--(that is <math>A^* = A</math> in the complex case and <math>A^T = A</math> in the real case)--> and <math>a</math>, <math>d</math> and <math>\det(A) = ad - bc</math> are (strictly) positive real numbers. <!-- Note that the parentheses convention doesn't really work here, it's the other way round.-->
 
{{off-topic|date=October 2013}}
 
Let the [[Banach space]]s <math>X</math> and <math>Y</math> be [[ordered vector space]]s and let <math>T \colon X \to Y</math> be a [[linear operator]].
The operator <math>T</math> is called ''positive'' if <math>Tx \geq 0</math> for all <math>x \geq 0</math> in <math>X</math>. For a positive operator <math>T</math> we write <math>T \geq 0</math>.
 
A positive operator maps the [[ordered vector space|positive cone]] of <math>X</math> onto a subset of the positive cone of <math>Y</math>. If <math>Y</math> is a [[field (mathematics)|field]] then <math>T</math> is called a positive [[linear functional]].
 
Many important operators are positive. For example:
* the [[Laplace operator]]s <math>-\Delta</math> and <math>-\frac{d^2}{dx^2}</math> are positive,
* the limit and [[Banach limit]] functionals are positive,
* the identity and absolute value operators are positive,
* the integral operator with a positive measure is positive.
 
The Laplace operator is an example of an unbounded positive linear operator. Hence, by the [[Hellinger-Toeplitz theorem]] it cannot be everywhere defined.
 
== Examples ==
* The following matrix <math>A</math> is not positive definite since <math>\det(A) = 0</math>. However, <math>A</math> is positive semidefinite since <math>a = 1</math>, <math>d = 1</math> and <math>\det(A) = 0</math> are non-negative.
:<math>A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}.</math>
 
== Partial ordering using positivity ==
<!--Hermitian elements are also called self-adjoint.--> <!--This has already been mentioned in the introduction.-->
By introducing the convention
:<math> A \geq B \iff A-B \text{ is positive}</math>
for self-adjoint elements in a C*-algebra <math>\mathcal{A}</math>, one obtains a [[partial order]] on the set of self-adjoint elements in <math>\mathcal{A}</math>. Note that according to this convention, we have <math>A \geq 0</math> if and only if <math>A</math> is positive, which is convenient.
 
This partial order is analoguous to the natural order on the real numbers, but only to some extent. For example, it respects multiplication by positive reals and addition of self-adjoint elements, but <math>AB \geq CD</math> need not hold for positive elements <math>A,B,C,D \in \mathcal{A}</math> with <math>A \geq C</math> and <math>B \geq D</math>.
 
== References ==
* {{Citation | last1=Conway | first1=John | title=A course in functional analysis | publisher=[[Springer Verlag]] | isbn=0-387-97245-5 | year=1990}}
 
{{DEFAULTSORT:Positive Element}}
[[Category:Functional analysis]]

Latest revision as of 09:24, 30 June 2014


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