In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groupss. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.

Definition

. A positive-definite function that satisfies

).

is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure with compact support.

Examples

, is positive-definite.

be a finite abelian group. Any character is positive-definite. (This is a special case of unitary representation.)

with the Lebesgue measure with compact support.

Unitary representations

A unitary representation.

in a natural way.

. This can be shown readily:

.

.

.

The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:

denotes the closure of the linear span.

.

Toeplitz kernels

is called a kernel of Toeplitz type, by analogy with Toeplitz matrices is a bounded operator is a contraction. This is precisely Sz.-Nagy's dilation theorem and hints at an important dilation-theoretic characterization of positivity that leads to a parametrization of arbitrary positive-definite kernels.

References