Positive element

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

If $T$ is a bounded linear operator on a complex Hilbert space $H$, then this notion coincides with the condition that $\langle Tx,x \rangle$ is non-negative for every vector $x$ in $H$. Note that $\langle Tx,x \rangle$ is real for every $x$ in $H$ if and only if $T$ is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint (and a self-adjoint everywhere defined operator on a Hilbert space is always bounded because of the Hellinger-Toeplitz theorem).

The set of positive elements of a C*-algebra forms a convex cone.

Positive and positive definite operators

A bounded linear operator $P$ on an inner product space $V$ is said to be positive (or positive semidefinite) if $P = S^*S$ for some bounded operator $S$ on $V$, and is said to be positive definite if $S$ is also non-singular.

(I) The following conditions for a bounded operator $P$ on $V$ to be positive semidefinite are equivalent:

• $P = S^*S$ for some bounded operator $S$ on $V$,
• $P = T^2$ for some self-adjoint operator $T$ on $V$,
• $P$ is self-adjoint and $\langle Pu,u \rangle \geq 0, \forall u \in V$.

(II) The following conditions for a bounded operator $P$ on $V$ to be positive definite are equivalent:

• $P = S^*S$ for some non-singular bounded operator $S$ on $V$,
• $P = T^2$ for some non-singular self-adjoint operator $T$ on $V$,
• $P$ is self adjoint and $\langle Pu,u \rangle > 0, \forall u \neq 0$ in $V$.

(III) A complex matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ represents a positive (semi)definite operator if and only if $A$ is Hermitian (or self-adjoint) and $a$, $d$ and $\det(A) = ad - bc$ are (strictly) positive real numbers.

Let the Banach spaces $X$ and $Y$ be ordered vector spaces and let $T \colon X \to Y$ be a linear operator. The operator $T$ is called positive if $Tx \geq 0$ for all $x \geq 0$ in $X$. For a positive operator $T$ we write $T \geq 0$.

A positive operator maps the positive cone of $X$ onto a subset of the positive cone of $Y$. If $Y$ is the field of $X$ then $T$ is called a positive linear functional.

Many important operators are positive. For example:

• the Laplace operators $-\Delta$ and $-\frac{d^2}{dx^2}$ 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 $A$ is not positive definite since $\det(A) = 0$. However, $A$ is positive semidefinite since $a = 1$, $d = 1$ and $\det(A) = 0$ are non-negative.

$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}.$

Partial ordering using positivity

By introducing the convention

$A \geq B \iff A-B \text{ is positive}$

for self-adjoint elements in a C*-algebra $\mathcal{A}$, one obtains a partial order on the set of self-adjoint elements in $\mathcal{A}$. Note that according to this convention, we have $A \geq 0$ if and only if $A$ 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 $AB \geq CD$ need not hold for positive elements $A,B,C,D \in \mathcal{A}$ with $A \geq C$ and $B \geq D$.