Convolution semigroups on hypergroups
For convolution semigroups \(\mu =(\mu_ t)_{t>0}\) on abelian hypergroups K a representation theorem is derived which extends earlier results of this (``Levy-Hinčin'') type: Roughly speaking a [abelian] hypergroup is a locally compact K with involution \({\;}^-\) on K and abstract convolution * on the bounded complex Borel measures on K which is associative [commutative] with \(p_ x*p_ y(\bar M)=p_{\bar x}*p_{\bar y}(M)\) and a unit \(p_ e\), \(p_ x:=\) Dirac measure in x, supp \(p_ x*p_ y\) compact and continuous is x, y, containing e iff \(x=\bar y\). The dual \(\hat K\) contains the bounded characters, i.e. \(\omega\) : \(K\to {\mathbb{C}}\) continuous with \(\omega(x)\omega(y)=\int \omega d(p_ x*p_ y)=:p_ x*p_ y(\omega)\) for \(x,y\in K\), with the topology of uniform convergence on compacta; here it is assumed that \(\hat K\) is again a hypergroup under the canonical product. A \(\mu: (0,\infty)\to\) nonnegative bounded Borel measures on K is called a convolution semigroup if \(\mu_ t(K)\leq 1\), \(\mu_ s*\mu_ t=\mu_{s+t}\) for \(s,t>0\) and \(\mu_ t\to p_ e\) if \(t\downarrow 0\). To such a \(\mu\) there exists a unique negative definite \(\phi: K\to {\mathbb{C}}\) with Re\(\phi\geq 0\) and \({\hat\mu}{}_ t(\omega)=\exp (- t\phi(\omega))\), \(\omega\in \hat K\), \(t>0\), the ``associated'' \(\phi\) ; \(\phi\) negative definite means \[ \sum_{j,k}(\phi (x_ j)+\overline{\phi(x_ k)}-p_{x_ j}*p_{x_ k}(\phi))c_ j\overline{c_ k}\geq 0 \] for all \(x_ 1,...,x_ n\in K\), \(c_ j\in {\mathbb{C}}\); examples of negative definite functions are \(\phi=ih\), \(h: K\to {\mathbb{R}}\) with \(p_ x*p_ y(h)=h(x)+h(y)\) and \(h(\bar x)=-h(x)\), or quadratic forms \(q>0\) (i.e. q: \(K\to {\mathbb{R}}\) with \(p_ x*p_{\bar y}(q)+p_ x*p_ y(q)=2(q(x)+q(y)))\). Under an additional assumption (F) on compact sets of \(\hat K\) the following Levy-Hinčin representation formula for convolution semigroups with symmetric Levy measure \(\lambda:=\lim_{t\downarrow 0}(1/t)\mu_ t\) is obtained, for the associated \(\phi\) : \[ \phi (\omega)=c+ih(\omega)+q(\omega)+\int_{K\setminus \{e\}}(1-Re \omega (x))d\lambda (x),\quad \omega \in \hat K, \] with real \(c\geq 0\), homomorphism h: \(\hat K\to {\mathbb{R}}\) as above and nonnegative quadratic form q: \(\hat K\to {\mathbb{R}}\), which are all unique. Special K subsumed and discussed are: Locally compact abelian groups, \({\mathbb{N}}_ 0=\{0,1,...\}\) with * induced by certain orthogonal polynomials, f.e. ultraspherical or Jacobi, or Gelfand pairs (G,H) with \(K=G//H=\{HgH:\) \(g\in G\}\) which for H subgroup of G generally does not inherit a multiplication, or \(K=G_ B=\bar B\)-orbit space in \(G=\) locally compact group, B some subgroup of automorphisms of G with relative compact orbits, B-invariant \(\mu_ t\); for \(G={\mathbb{R}}^ n\), \(B=SO(n)\), the \(\omega\in \hat G_ B\) are given by Bessel functions \(J_{-1+n/2}\).
scientific article; zbMATH DE number 4063760