Positive-definite function on a group explained

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 groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.

Definition

Let

be a group,

be a complex Hilbert space, and

L(H)

be the bounded operators on

. A positive-definite function on

is a function

F:G\toL(H)

that satisfies

\sum_s,t\langleF(s^-1t)h(t),h(s)\rangle\geq0,

for every function

h:G\toH

with finite support (

takes non-zero values for only finitely many

In other words, a function

F:G\toL(H)

is said to be a positive-definite function if the kernel

K:G x G\toL(H)

defined by

K(s,t)=F(s^-1t)

is a positive-definite kernel. Such a kernel is

-symmetric, that is, it invariant under left

-action:

K(s, t) = K(rs, rt), \quad \forall r \in G

When

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

\mu

. A positive-definite function on

is a continuous function

F:G\toL(H)

that satisfies

\int_\langle F(s^t) h(t), h(s) \rangle \; \mu(ds) \mu(dt) \geq 0,

for every continuous function

h:G\toH

with compact support.

Examples

The constant function

F(g)=I

, where

is the identity operator on

, is positive-definite.

Let

be a finite abelian group and

be the one-dimensional Hilbert space

. Any character

\chi:G\toC

is positive-definite. (This is a special case of unitary representation.)

To show this, recall that a character of a finite group

is a homomorphism from

to the multiplicative group of norm-1 complex numbers. Then, for any function

h:G\toC

\sum_\chi(s^t)h(t)\overline = \sum_\chi(s^)h(t)\chi(t)\overline= \sum_\chi(s^)\overline\sum_h(t)\chi(t) = \left|\sum_h(t)\chi(t)\right|^2 \geq 0.

When

G=\Rⁿ

with the Lebesgue measure, and

H=\C^m

, a positive-definite function on

is a continuous function

F:\Rⁿ\to\C^{m x}

such that

\int_ h(x)^\dagger F(x-y) h(y)\; dxdy \geq 0

for every continuous function

h:\Rⁿ\to\C^m

with compact support.

Unitary representations

A unitary representation is a unital homomorphism

\Phi:G\toL(H)

where

\Phi(s)

is a unitary operator for all

. For such

\Phi

\Phi(s^-1)=\Phi(s)^*

Positive-definite functions on

are intimately related to unitary representations of

. Every unitary representation of

gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of

in a natural way.

Let

\Phi:G\toL(H)

be a unitary representation of

. If

P\inL(H)

is the projection onto a closed subspace

. Then

F(s)=P\Phi(s)

is a positive-definite function on

with values in

L(H')

. This can be shown readily:

\begin{align} \sum_s,t\langleF(s^-1t)h(t),h(s)\rangle&=\sum_s,t\langleP\Phi(s^-1t)h(t),h(s)\rangle\ {}&=\sum_s,t\langle\Phi(t)h(t),\Phi(s)h(s)\rangle\ {}&=\left\langle\sum_t\Phi(t)h(t),\sum_s\Phi(s)h(s)\right\rangle\ {}&\geq0 \end{align}

for every

h:G\toH'

with finite support. If

has a topology and

\Phi

is weakly(resp. strongly) continuous, then clearly so is

On the other hand, consider now a positive-definite function

. A unitary representation of

can be obtained as follows. Let

C₀₀(G,H)

be the family of functions

h:G\toH

with finite support. The corresponding positive kernel

K(s,t)=F(s^-1t)

defines a (possibly degenerate) inner product on

C₀₀(G,H)

. Let the resulting Hilbert space be denoted by

We notice that the "matrix elements"

K(s,t)=K(a^-1s,a^-1t)

for all

a,s,t

. So

U_ah(s)=h(a^-1s)

preserves the inner product on

, i.e. it is unitary in

L(V)

. It is clear that the map

\Phi(a)=U_a

is a representation of

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

V=vee_s\Phi(s)H

where

vee

denotes the closure of the linear span.

Identify

as elements (possibly equivalence classes) in

, whose support consists of the identity element

e\inG

, and let

be the projection onto this subspace. Then we have

PU_aP=F(a)

for all

a\inG

Toeplitz kernels

Let

be the additive group of integers

. The kernel

K(n,m)=F(m-n)

is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If

is of the form

F(n)=Tⁿ

where

is a bounded operator acting on some Hilbert space. One can show that the kernel

K(n,m)

is positive if and only if

is a contraction. By the discussion from the previous section, we have a unitary representation of

\Phi(n)=Uⁿ

for a unitary operator

. Moreover, the property

PU_aP=F(a)

now translates to

PU^nP=Tⁿ

. 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

Book: Berg . Christian . Christensen . Paul . Ressel . Harmonic Analysis on Semigroups . Graduate Texts in Mathematics . 100 . Springer Verlag . 1984.
Book: Constantinescu, T. . Schur Parameters, Dilation and Factorization Problems . Birkhauser Verlag . 1996.
Book: Sz.-Nagy . B. . Foias . C. . Harmonic Analysis of Operators on Hilbert Space . North-Holland . 1970.
Book: Sasvári, Z. . Positive Definite and Definitizable Functions . Akademie Verlag . 1994.
Book: Wells . J. H. . Williams . L. R. . Embeddings and extensions in analysis . Ergebnisse der Mathematik und ihrer Grenzgebiete . 84 . Springer-Verlag . New York-Heidelberg . 1975 . vii+108.