Positive operator (Hilbert space)

In mathematics, especially functional analysis, a hermitian element A of a C*-algebra is a positive element if its spectrum consists of positive real numbers. More strongly, an element x of a C*-algebra A is positive if and only if there is some b in A such that x = b^*b. A positive element is self-adjoint and thus normal.

If A is a bounded linear operator on a Hilbert space H, then this notion coincides with the condition that A is self-adjoint and $\langle Ax,x\rangle$ is nonnegative for every vector x in H. Note that $\langle Ax,x\rangle$ is real for every x in H since A is self-adjoint.

The set of positive elements form a convex cone.

Positive and Positive Definite Operators

A linear operator $P$ on an inner product space $V$ is said to be positive [or semidefinite] if $P=S*S$ for some operator $S$ and is said to be positive definite if $S$ is also nonsingular.

Relevant theorems:

(I) The following conditions on an operator P are equivalent:

• $P=T^{2}$ for some self-adjoint operator T.
• P=S*S for some operator S.
• P is self adjoint and $\left\langle P(u),u\right\rangle \geq 0$ $\forall u\in V$.

(II) The following conditions on an operator P are equivalent:

• $P=T^{2}$ for some nonsingular self-adjoint operator T.
• $P=S*S$ for some nonsingular operator S.
• P is self adjoint and $\left\langle P(u),u\right\rangle >0$ $\forall u\neq 0$ in V.

(III) A complex matrix $A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}.$ represents a positive (positive definite) operator if and only if A is self-adjoint (that is $A*A$ in the complex case and $A^{T}=A$ in the real case) and a,d and |A|=ad-bc are nonnegative (positive) real numbers.

Let the Banach spaces $X,Y$ be ordered vector spaces and the operator $T:X\to Y$ a linear map. The operator $T$ is called positive (written $T\geq 0$) if $Tx\geq 0$ whenever $x\geq 0$.

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

The Laplace operator is an example of an unbounded positive linear operator.

Examples

• The following matrix is not positive definite, since |A|=0. However A is positive since a=1, d=1 and |A|=0 are nonnegative:

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

Partial ordering using positivity

Hermitian elements are also called self-adjoint. By introducing the convention

$A\leq B\Longleftrightarrow B-A\ {\textrm {is}}\ {\textrm {positive}}$

for self-adjoint elements in a C*-algebra ${\mathcal {A}}$, one obtains a partial ordering of ${\mathcal {A}}$.

This partial ordering is analoguous to the ordering of real numbers, but only to some extent. For example, it respects multiplication by positive reals and addition with positive elements, but $AB\geq CD$ need not hold for positive elements $A,B,C,D\in {\mathcal {A}}$ with $A\geq C$ and $B\geq D$.

References

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