Positive element

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

If ${\displaystyle T}$ is a bounded linear operator on a complex Hilbert space ${\displaystyle H}$, then this notion coincides with the condition that ${\displaystyle \langle Tx,x\rangle }$ is non-negative for every vector ${\displaystyle x}$ in ${\displaystyle H}$. Note that ${\displaystyle \langle Tx,x\rangle }$ is real for every ${\displaystyle x}$ in ${\displaystyle H}$ if and only if ${\displaystyle T}$ 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 ${\displaystyle P}$ on an inner product space ${\displaystyle V}$ is said to be positive (or positive semidefinite) if ${\displaystyle P=S^{*}S}$ for some bounded operator ${\displaystyle S}$ on ${\displaystyle V}$, and is said to be positive definite if ${\displaystyle S}$ is also non-singular.

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

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

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

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

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

Examples

• The following matrix ${\displaystyle A}$ is not positive definite since ${\displaystyle \det(A)=0}$. However, ${\displaystyle A}$ is positive semidefinite since ${\displaystyle a=1}$, ${\displaystyle d=1}$ and ${\displaystyle \det(A)=0}$ are non-negative.

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

Partial ordering using positivity

By defining

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

for self-adjoint elements in a C*-algebra ${\displaystyle {\mathcal {A}}}$, one obtains a partial order on the set of self-adjoint elements in ${\displaystyle {\mathcal {A}}}$. Note that according to this definition, we have ${\displaystyle A\geq 0}$ if and only if ${\displaystyle A}$ is positive, which is convenient.

This partial order is analogous 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 ${\displaystyle AB\geq CD}$ need not hold for positive elements ${\displaystyle A,B,C,D\in {\mathcal {A}}}$ with ${\displaystyle A\geq C}$ and ${\displaystyle B\geq D}$.

References

• Conway, John (1990), A course in functional analysis, Springer Verlag, ISBN 0-387-97245-5