Positive definite function on a group

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.


Let "G" be a group, "H" be a complex Hilbert space, and "L"("H") be the bounded operators on "H". A positive definite function on "G" is a function "F": "G" → "L"("H") that satisfies

:sum_{s,t in G}langle F(s^{-1}t) h(t), h(s) angle geq 0 ,

for every function "h": "G" → "H" with finite support ("h" takes non-zero values for only finitely many "s").

In other words, a function "F": "G" → "L"("H") is said to be a positive function if the kernel "K": "G" × "G" → "L"("H") defined by "K"("s", "t") = "F"("s"-1"t") is a positive definite kernel.

Unitary representations

An unitary representation is an unital homomorphism Φ: "G" → "L"("H") where Φ("s") is an unitary operator for all "s". For such Φ, Φ("s"-1) = Φ("s")*.

Positive functions on "G" is intimately related to unitary representations of "G". Every unitary representation of "G" gives rise to a family of positive definite functions. Conversely, given a positive definite function, one can define a unitary representation of "G" in a natural way.

Let Φ: "G" → "L"("H") be a unitary representation of "G". If "P" ∈ "L"("H") is the projection onto a closed subspace "H`" of "H". Then "F"("s") = "P" Φ("s") is a positive definite function on "G" with values in "L"("H`"). This can be shown readily:

:egin{array}{rl}sum_{s,t in G}langle F(s^{-1}t) h(t), h(s) angle & =sum_{s,t in G}langle P Phi (s^{-1}t) h(t), h(s) angle \ {} & =sum_{s,t in G}langle Phi (t) h(t), Phi(s)h(s) angle \ {} & = langle sum_{t in G} Phi (t) h(t), sum_{s in G} Phi(s)h(s) angle \ {} & geq 0end{array}

for every "h": "G" → "H`" with finite support. If "G" has a topology and Φ is weakly(resp. strongly) continuous, then clearly so is "F".

On the other hand, consider now a positive definite function "F" on "G". An unitary representation of "G" can be obtained as follows. Let "C"00("G", "H") be the family of functions "h": "G" → "H" with finite support. The corresponding positive kernel "K"("s", "t") = "F"("s"-1"t") defines a (possibly degenerate) inner product on "C"00("G", "H"). Let the resulting Hilbert space be denoted by "V".

We notice that the "matrix elements" "K"("s", "t") = "K"("a"-1"s", "a"-1"t") for all "a", "s", "t" in "G". So "Uah"("s") = "h"("a"-1"s") preserves the inner product on "V", i.e. it is unitary in "L"("V"). It is clear that the map Φ("a") = "U"a is a representation of "G" on "V".

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

:V = igvee_{s in G} Phi(s)H

where igvee denotes the closure of the linear span.

Identify "H" as, elements (possibly equivalence classes) in "V", whose support consists of the identity element "e" ∈ "G", and let "P" be the projection onto this subspace. Then we have "PUaP" = "F"("a") for all "a" ∈ "G".

Toeplitz kernels

Let "G" be the additive group of integers Z. The kernel "K"("n", "m") = "F"("m" - "n") is called a kernel of "Toeplitz" type, by analogy with Toeplitz matrices. If "F" is of the form "F"("n") = "Tn" where "T" is a bounded operator acting on some Hilbert space. One can show that the kernel "K"("n", "m") is positive if and only if "T" is a contraction. By the discussion from the previous section, we have a unitary representation of Z, Φ("n") = "U""n" for an unitary operator "U". Moreover, the property "PUaP" = "F"("a") now translates to "PUnP" = "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.


*T. Constantinescu, "Schur Parameters, Dilation and Factorization Problems", Birkhauser Verlag, 1996.
*B. Sz.-Nagy and C. Foias, "Harmonic Analysis of Operators on Hilbert Space," North-Holland, 1970.
*Z. Sasvári, "Positive Definite and Definitizable Functions", Akademie Verlag, 1994

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Positive-definite function — In mathematics, the term positive definite function may refer to a couple of different concepts. Contents 1 In dynamical systems 2 In analysis 2.1 Bochner s theorem 2.1.1 Applications …   Wikipedia

  • Positive definite — In mathematics, positive definite may refer to: * positive definite matrix * positive definite function ** positive definite function on a group * positive definite bilinear form …   Wikipedia

  • Positive definite kernel — In operator theory, a positive definite kernel is a generalization of a positive matrix. Definition Let :{ H n } {n in {mathbb Z be a sequence of (complex) Hilbert spaces and :mathcal{L}(H i, H j)be the bounded operators from Hi to Hj . A map A… …   Wikipedia

  • Function (mathematics) — f(x) redirects here. For the band, see f(x) (band). Graph of example function, In mathematics, a function associates one quantity, the a …   Wikipedia

  • Zonal spherical function — In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K invariant vector in an… …   Wikipedia

  • Theta function — heta 1 with u = i pi z and with nome q = e^{i pi au}= 0.1 e^{0.1 i pi}. Conventions are (mathematica): heta 1(u;q) = 2 q^{1/4} sum {n=0}^infty ( 1)^n q^{n(n+1)} sin((2n+1)u) this is: heta 1(u;q) = sum {n= infty}^{n=infty} ( 1)^{n 1/2}… …   Wikipedia

  • Gamma function — For the gamma function of ordinals, see Veblen function. The gamma function along part of the real axis In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its… …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Bochner's theorem — In mathematics, Bochner s theorem characterizes the Fourier transform of a positive finite Borel measure on the real line. Background Given a positive finite Borel measure μ on the real line R, the Fourier transform Q of μ is the continuous… …   Wikipedia

  • List of harmonic analysis topics — This is a list of harmonic analysis topics, by Wikipedia page. See also list of Fourier analysis topics and list of Fourier related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.