Positive Element - Positive and Positive Definite Operators

Positive and Positive Definite Operators

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 ,
• 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 spaces and be ordered vector spaces 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 of onto a subset of the positive cone of . If is a field then is called a positive linear functional.

Many important operators are positive. For example:

• the Laplace operators 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.