Positive element

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In mathematics, especially functional analysis, a self-adjoint (or Hermitian) 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 linear operator on a complex Hilbert space , 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 (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[edit]

A bounded linear operator on an inner product space 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 ,
  • .

(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 ,
  • 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.

Examples[edit]

  • The following matrix is not positive definite since . However, is positive semidefinite since , and are non-negative.

Partial ordering using positivity[edit]

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 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 need not hold for positive elements with and .

References[edit]

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