Positive-definite function on a group

☆ Save On Wikipedia ↗

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 G {\displaystyle G} {\displaystyle G} be a group, H {\displaystyle H} {\displaystyle H} be a complex Hilbert space, and L ( H ) {\displaystyle L(H)} {\displaystyle L(H)} be the bounded operators on H {\displaystyle H} {\displaystyle H}. A positive-definite function on G {\displaystyle G} {\displaystyle G} is a function F : G → L ( H ) {\displaystyle F:G\to L(H)} {\displaystyle F:G\to L(H)} that satisfies

∑ s , t ∈ G ⟨ F ( s − 1 t ) h ( t ) , h ( s ) ⟩ ≥ 0 , {\displaystyle \sum _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \geq 0,} {\displaystyle \sum _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \geq 0,}

for every function h : G → H {\displaystyle h:G\to H} {\displaystyle h:G\to H} with finite support ( h {\displaystyle h} {\displaystyle h} takes non-zero values for only finitely many s {\displaystyle s} {\displaystyle s}).

In other words, a function F : G → L ( H ) {\displaystyle F:G\to L(H)} {\displaystyle F:G\to L(H)} is said to be a positive-definite function if the kernel K : G × G → L ( H ) {\displaystyle K:G\times G\to L(H)} {\displaystyle K:G\times G\to L(H)} defined by K ( s , t ) = F ( s − 1 t ) {\displaystyle K(s,t)=F(s^{-1}t)} {\displaystyle K(s,t)=F(s^{-1}t)} is a positive-definite kernel. Such a kernel is G {\displaystyle G} {\displaystyle G}-symmetric, that is, it invariant under left G {\displaystyle G} {\displaystyle G}-action: K ( s , t ) = K ( r s , r t ) , ∀ r ∈ G {\displaystyle K(s,t)=K(rs,rt),\quad \forall r\in G} {\displaystyle K(s,t)=K(rs,rt),\quad \forall r\in G}When G {\displaystyle G} {\displaystyle G} is a locally compact group, the definition generalizes by integration over its left-invariant Haar measure μ {\displaystyle \mu } {\displaystyle \mu }. A positive-definite function on G {\displaystyle G} {\displaystyle G} is a continuous function F : G → L ( H ) {\displaystyle F:G\to L(H)} {\displaystyle F:G\to L(H)} that satisfies ∫ s , t ∈ G ⟨ F ( s − 1 t ) h ( t ) , h ( s ) ⟩ μ ( d s ) μ ( d t ) ≥ 0 , {\displaystyle \int _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \;\mu (ds)\mu (dt)\geq 0,} {\displaystyle \int _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle \;\mu (ds)\mu (dt)\geq 0,}for every continuous function h : G → H {\displaystyle h:G\to H} {\displaystyle h:G\to H} with compact support.

Examples

The constant function F ( g ) = I {\displaystyle F(g)=I} {\displaystyle F(g)=I}, where I {\displaystyle I} {\displaystyle I} is the identity operator on H {\displaystyle H} {\displaystyle H}, is positive-definite.

Let G {\displaystyle G} {\displaystyle G} be a finite abelian group and H {\displaystyle H} {\displaystyle H} be the one-dimensional Hilbert space C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }. Any character χ : G → C {\displaystyle \chi :G\to \mathbb {C} } {\displaystyle \chi :G\to \mathbb {C} } is positive-definite. (This is a special case of unitary representation.)

To show this, recall that a character of a finite group G {\displaystyle G} {\displaystyle G} is a homomorphism from G {\displaystyle G} {\displaystyle G} to the multiplicative group of norm-1 complex numbers. Then, for any function h : G → C {\displaystyle h:G\to \mathbb {C} } {\displaystyle h:G\to \mathbb {C} }, ∑ s , t ∈ G χ ( s − 1 t ) h ( t ) h ( s ) ¯ = ∑ s , t ∈ G χ ( s − 1 ) h ( t ) χ ( t ) h ( s ) ¯ = ∑ s χ ( s − 1 ) h ( s ) ¯ ∑ t h ( t ) χ ( t ) = | ∑ t h ( t ) χ ( t ) | 2 ≥ 0. {\displaystyle \sum _{s,t\in G}\chi (s^{-1}t)h(t){\overline {h(s)}}=\sum _{s,t\in G}\chi (s^{-1})h(t)\chi (t){\overline {h(s)}}=\sum _{s}\chi (s^{-1}){\overline {h(s)}}\sum _{t}h(t)\chi (t)=\left|\sum _{t}h(t)\chi (t)\right|^{2}\geq 0.} {\displaystyle \sum _{s,t\in G}\chi (s^{-1}t)h(t){\overline {h(s)}}=\sum _{s,t\in G}\chi (s^{-1})h(t)\chi (t){\overline {h(s)}}=\sum _{s}\chi (s^{-1}){\overline {h(s)}}\sum _{t}h(t)\chi (t)=\left|\sum _{t}h(t)\chi (t)\right|^{2}\geq 0.}When G = R n {\displaystyle G=\mathbb {R} ^{n}} {\displaystyle G=\mathbb {R} ^{n}} with the Lebesgue measure, and H = C m {\displaystyle H=\mathbb {C} ^{m}} {\displaystyle H=\mathbb {C} ^{m}}, a positive-definite function on G {\displaystyle G} {\displaystyle G} is a continuous function F : R n → C m × m {\displaystyle F:\mathbb {R} ^{n}\to \mathbb {C} ^{m\times m}} {\displaystyle F:\mathbb {R} ^{n}\to \mathbb {C} ^{m\times m}} such that ∫ x , y ∈ R n h ( x ) † F ( x − y ) h ( y ) d x d y ≥ 0 {\displaystyle \int _{x,y\in \mathbb {R} ^{n}}h(x)^{\dagger }F(x-y)h(y)\;dxdy\geq 0} {\displaystyle \int _{x,y\in \mathbb {R} ^{n}}h(x)^{\dagger }F(x-y)h(y)\;dxdy\geq 0}for every continuous function h : R n → C m {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {C} ^{m}} {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {C} ^{m}} with compact support.

Unitary representations

A unitary representation is a unital homomorphism Φ : G → L ( H ) {\displaystyle \Phi :G\to L(H)} {\displaystyle \Phi :G\to L(H)} where Φ ( s ) {\displaystyle \Phi (s)} {\displaystyle \Phi (s)} is a unitary operator for all s {\displaystyle s} {\displaystyle s}. For such Φ {\displaystyle \Phi } {\displaystyle \Phi }, Φ ( s − 1 ) = Φ ( s ) ∗ {\displaystyle \Phi (s^{-1})=\Phi (s)^{*}} {\displaystyle \Phi (s^{-1})=\Phi (s)^{*}}.

Positive-definite functions on G {\displaystyle G} {\displaystyle G} are intimately related to unitary representations of G {\displaystyle G} {\displaystyle G}. Every unitary representation of G {\displaystyle G} {\displaystyle G} gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of G {\displaystyle G} {\displaystyle G} in a natural way.

Let Φ : G → L ( H ) {\displaystyle \Phi :G\to L(H)} {\displaystyle \Phi :G\to L(H)} be a unitary representation of G {\displaystyle G} {\displaystyle G}. If P ∈ L ( H ) {\displaystyle P\in L(H)} {\displaystyle P\in L(H)} is the projection onto a closed subspace H ′ {\displaystyle H'} {\displaystyle H'} of H {\displaystyle H} {\displaystyle H}. Then F ( s ) = P Φ ( s ) {\displaystyle F(s)=P\Phi (s)} {\displaystyle F(s)=P\Phi (s)} is a positive-definite function on G {\displaystyle G} {\displaystyle G} with values in L ( H ′ ) {\displaystyle L(H')} {\displaystyle L(H')}. This can be shown readily:

∑ s , t ∈ G ⟨ F ( s − 1 t ) h ( t ) , h ( s ) ⟩ = ∑ s , t ∈ G ⟨ P Φ ( s − 1 t ) h ( t ) , h ( s ) ⟩ = ∑ s , t ∈ G ⟨ Φ ( t ) h ( t ) , Φ ( s ) h ( s ) ⟩ = ⟨ ∑ t ∈ G Φ ( t ) h ( t ) , ∑ s ∈ G Φ ( s ) h ( s ) ⟩ ≥ 0 {\displaystyle {\begin{aligned}\sum _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle &=\sum _{s,t\in G}\langle P\Phi (s^{-1}t)h(t),h(s)\rangle \\{}&=\sum _{s,t\in G}\langle \Phi (t)h(t),\Phi (s)h(s)\rangle \\{}&=\left\langle \sum _{t\in G}\Phi (t)h(t),\sum _{s\in G}\Phi (s)h(s)\right\rangle \\{}&\geq 0\end{aligned}}} {\displaystyle {\begin{aligned}\sum _{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle &=\sum _{s,t\in G}\langle P\Phi (s^{-1}t)h(t),h(s)\rangle \\{}&=\sum _{s,t\in G}\langle \Phi (t)h(t),\Phi (s)h(s)\rangle \\{}&=\left\langle \sum _{t\in G}\Phi (t)h(t),\sum _{s\in G}\Phi (s)h(s)\right\rangle \\{}&\geq 0\end{aligned}}}

for every h : G → H ′ {\displaystyle h:G\to H'} {\displaystyle h:G\to H'} with finite support. If G {\displaystyle G} {\displaystyle G} has a topology and Φ {\displaystyle \Phi } {\displaystyle \Phi } is weakly(resp. strongly) continuous, then clearly so is F {\displaystyle F} {\displaystyle F}.

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

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

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

V = ⋁ s ∈ G Φ ( s ) H {\displaystyle V=\bigvee _{s\in G}\Phi (s)H} {\displaystyle V=\bigvee _{s\in G}\Phi (s)H}

where ⋁ {\displaystyle \bigvee } {\displaystyle \bigvee } denotes the closure of the linear span.

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

Toeplitz kernels

Let G {\displaystyle G} {\displaystyle G} be the additive group of integers Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }. The kernel K ( n , m ) = F ( m − n ) {\displaystyle K(n,m)=F(m-n)} {\displaystyle K(n,m)=F(m-n)} is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If F {\displaystyle F} {\displaystyle F} is of the form F ( n ) = T n {\displaystyle F(n)=T^{n}} {\displaystyle F(n)=T^{n}} where T {\displaystyle T} {\displaystyle T} is a bounded operator acting on some Hilbert space, one can show that the kernel K ( n , m ) {\displaystyle K(n,m)} {\displaystyle K(n,m)} is positive if and only if T {\displaystyle T} {\displaystyle T} is a contraction. By the discussion from the previous section, we have a unitary representation of Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }, Φ ( n ) = U n {\displaystyle \Phi (n)=U^{n}} {\displaystyle \Phi (n)=U^{n}} for a unitary operator U {\displaystyle U} {\displaystyle U}. Moreover, the property P U a P = F ( a ) {\displaystyle PU_{a}P=F(a)} {\displaystyle PU_{a}P=F(a)} now translates to P U n P = T n {\displaystyle PU^{n}P=T^{n}} {\displaystyle PU^{n}P=T^{n}}. 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

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