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Positive-definite function on a group
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<h2 id="definition">Definition</h2> <p>Let G {\displaystyle G} be a group, H {\displaystyle H} be a complex Hilbert space, and L ( H ) {\displaystyle L(H)} be the bounded operators on H {\displaystyle H} . A positive-definite function on G {\displaystyle G} is a function F : G → L ( H ) {\displaystyle F:G\to L(H)} that satisfies </p> ∑ 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,} <p>for every function h : G → H {\displaystyle h:G\to H} with finite support ( h {\displaystyle h} takes non-zero values for only finitely many s {\displaystyle s} ). </p><p>In other words, a function F : G → 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)} defined by 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} -symmetric, that is, it invariant under left 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} When G {\displaystyle G} is a <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact group</a>, the definition generalizes by integration over its left-invariant <a href="/facts/Haar_measure/2IcmTcuU">Haar measure</a> μ {\displaystyle \mu } . A positive-definite function on G {\displaystyle G} is a continuous function F : G → 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,} for every continuous function h : G → H {\displaystyle h:G\to H} with <a href="/facts/Compact_support/BTuMGbsb">compact support</a>. </p> <h3>Examples</h3> <p>The constant function F ( g ) = I {\displaystyle F(g)=I} , where I {\displaystyle I} is the identity operator on H {\displaystyle H} , is positive-definite. </p><p>Let G {\displaystyle G} be a finite abelian group and H {\displaystyle H} be the one-dimensional Hilbert space C {\displaystyle \mathbb {C} } . Any <a href="/facts/Character_theory/WMp5Q6YE">character</a> χ : G → C {\displaystyle \chi :G\to \mathbb {C} } is positive-definite. (This is a special case of unitary representation.) </p><p>To show this, recall that a character of a finite group G {\displaystyle G} is a homomorphism from 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} } , ∑ 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.} When G = R n {\displaystyle G=\mathbb {R} ^{n}} with the <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>, and H = C m {\displaystyle H=\mathbb {C} ^{m}} , a positive-definite function on G {\displaystyle G} is a continuous function F : R n → C m × 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} for every continuous function h : R n → C m {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {C} ^{m}} with compact support. </p> <h2 id="unitary-representations">Unitary representations</h2> <p>A <a href="/facts/Unitary_representation/W1XRatG8">unitary representation</a> is a unital homomorphism Φ : G → L ( H ) {\displaystyle \Phi :G\to L(H)} where Φ ( s ) {\displaystyle \Phi (s)} is a unitary operator for all s {\displaystyle s} . For such Φ {\displaystyle \Phi } , Φ ( s − 1 ) = Φ ( s ) ∗ {\displaystyle \Phi (s^{-1})=\Phi (s)^{*}} . </p><p>Positive-definite functions on G {\displaystyle G} are intimately related to unitary representations of G {\displaystyle G} . Every unitary representation of 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} in a natural way. </p><p>Let Φ : G → L ( H ) {\displaystyle \Phi :G\to L(H)} be a unitary representation of G {\displaystyle G} . If P ∈ L ( H ) {\displaystyle P\in L(H)} is the projection onto a closed subspace H ′ {\displaystyle H'} of H {\displaystyle H} . Then F ( s ) = P Φ ( s ) {\displaystyle F(s)=P\Phi (s)} is a positive-definite function on G {\displaystyle G} with values in L ( H ′ ) {\displaystyle L(H')} . This can be shown readily: </p> ∑ 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}}} <p>for every h : G → H ′ {\displaystyle h:G\to H'} with finite support. If G {\displaystyle G} has a topology and Φ {\displaystyle \Phi } is weakly(resp. strongly) continuous, then clearly so is F {\displaystyle F} . </p><p>On the other hand, consider now a positive-definite function F {\displaystyle F} on G {\displaystyle G} . A unitary representation of G {\displaystyle G} can be obtained as follows. Let C 00 ( G , H ) {\displaystyle C_{00}(G,H)} be the family of functions h : G → 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)} defines a (possibly degenerate) inner product on C 00 ( G , H ) {\displaystyle C_{00}(G,H)} . Let the resulting Hilbert space be denoted by V {\displaystyle V} . </p><p>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)} for all a , s , t {\displaystyle a,s,t} in G {\displaystyle G} . So 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} , i.e. it is unitary in L ( V ) {\displaystyle L(V)} . It is clear that the map Φ ( a ) = U a {\displaystyle \Phi (a)=U_{a}} is a representation of G {\displaystyle G} on V {\displaystyle V} . </p><p>The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds: </p> V = ⋁ s ∈ G Φ ( s ) H {\displaystyle V=\bigvee _{s\in G}\Phi (s)H} <p>where ⋁ {\displaystyle \bigvee } denotes the closure of the linear span. </p><p>Identify H {\displaystyle H} as elements (possibly equivalence classes) in V {\displaystyle V} , whose support consists of the identity element e ∈ G {\displaystyle e\in G} , and let P {\displaystyle P} be the projection onto this subspace. Then we have P U a P = F ( a ) {\displaystyle PU_{a}P=F(a)} for all a ∈ G {\displaystyle a\in G} . </p> <h2 id="toeplitz-kernels">Toeplitz kernels</h2> <p>Let G {\displaystyle G} be the additive group of integers Z {\displaystyle \mathbb {Z} } . The kernel K ( n , m ) = F ( m − n ) {\displaystyle K(n,m)=F(m-n)} is called a kernel of <i>Toeplitz</i> type, by analogy with <a href="/facts/Toeplitz_matrix/WAjSdlaW">Toeplitz matrices</a>. If F {\displaystyle F} is of the form F ( n ) = T n {\displaystyle F(n)=T^{n}} where 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)} is positive if and only if T {\displaystyle T} is a <a href="/facts/Contraction_(operator_theory)/W4UcMlRd">contraction</a>. By the discussion from the previous section, we have a unitary representation of Z {\displaystyle \mathbb {Z} } , Φ ( n ) = U n {\displaystyle \Phi (n)=U^{n}} for a unitary operator U {\displaystyle U} . Moreover, the property P U 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}} . This is precisely <a href="/facts/Sz.-Nagy%27s_dilation_theorem/G5juefwd">Sz.-Nagy's dilation theorem</a> and hints at an important dilation-theoretic characterization of positivity that leads to a parametrization of arbitrary positive-definite kernels. </p> <ul><li>Berg, Christian; Christensen, Paul; Ressel (1984). <i>Harmonic Analysis on Semigroups</i>. Graduate Texts in Mathematics. Vol. 100. Springer Verlag.</li> <li>Constantinescu, T. (1996). <i>Schur Parameters, Dilation and Factorization Problems</i>. Birkhauser Verlag.</li> <li>Sz.-Nagy, B.; Foias, C. (1970). <i>Harmonic Analysis of Operators on Hilbert Space</i>. North-Holland.</li> <li>Sasvári, Z. (1994). <i>Positive Definite and Definitizable Functions</i>. Akademie Verlag.</li> <li>Wells, J. H.; Williams, L. R. (1975). <i>Embeddings and extensions in analysis</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 84. New York-Heidelberg: Springer-Verlag. pp. vii+108.</li></ul>