In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, a self-adjoint

Self-adjoint

In mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...

 (or hermitian

Hermitian

A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...

) element of a C*-algebra  is called positive if its spectrum  consists of non-negative real numbers. Moreover, an element of a C*-algebra is positive if and only if there is some in such that . A positive element is self-adjoint and thus normal.

If is a bounded

Bounded operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...

 linear operator on a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 , then this notion coincides with the condition that is non-negative for every vector in . Note that is real for every in if and only if ' is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint

Self-adjoint operator

In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

 (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

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

.

Positive and positive definite operators

A bounded linear operator on an inner product space

Inner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

  is said to be positive (or positive semidefinite) if for some bounded operator on , and is said to be positive definite if is also non-singular.
(I) The following conditions for a bounded operator on to be positive semidefinite are equivalent:

• for some bounded operator on ,
• for some self-adjoint operator on ,
• is self-adjoint and .

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

• for some non-singular bounded operator on ,
• for some non-singular self-adjoint operator on ,
• is self adjoint and in .

(III) A complex matrix represents a positive (semi)definite operator if and only if is hermitian (or self-adjoint) and , and are (strictly) positive real numbers.

Let the Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s and be ordered vector space

Ordered vector space

In mathematics an ordered vector space or partially ordered vector space is a vector space equipped with a partial order which is compatible with the vector space operations.- Definition:...

s and let be a linear operator.
The operator is called positive if for all in . For a positive operator we write .

A positive operator maps the positive cone

Ordered vector space

In mathematics an ordered vector space or partially ordered vector space is a vector space equipped with a partial order which is compatible with the vector space operations.- Definition:...

 of onto a subset of the positive cone of . If is a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 then is called a positive linear functional

Linear functional

In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

.

Many important operators are positive. For example:

• the Laplace operator
  Laplace operator
  In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
  
  s and 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 is not positive definite since . However, is positive semidefinite since , and are non-negative.

Partial ordering using positivity

By introducing the convention
for self-adjoint elements in a C*-algebra , one obtains a partial order on the set of self-adjoint elements in . Note that according to this convention, we have if and only if 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 need not hold for positive elements with and .