Positive definite kernels on the complex Hilbert sphere
Let H be an infinite-dimensional complex Hilbert space, and let L be the set of complex-valued functions f on \(D:=\{z\in {\mathbb{C}}:\) \(| z| \leq 1\}\) such that \((x,y)\mapsto f(<x,y>)\) is a positive definite kernel on the unit sphere S of H. The authors prove that a continuous function \(f\in L\) has a unique series representation \(f(z)=\sum a_{n,m} z^ n\bar z^ m\) (z\(\in D)\), where all \(a_{n,m}\geq 0\) and \(\sum a_{n,m}<\infty\). Following a hint of I. J. Schoenberg they identify \(L_ 1:=\{f\in L:\) \(f(1)=1\}\) with the set K of all functions \(f: D\to {\mathbb{C}}\) with \(f(1)=1\) and the following property: If \((a_{jk})\) is a positive semidefinite complex matrix then \((f(a_{jk}))\) is positive semidefinite. They show that K is a Bauer simplex whose set of extreme points is the closure of \(\{z^ n\bar z^ m:\) \(n,m\in {\mathbb{N}}_ 0\}\) in \({\mathbb{C}}^ D\), and establish their result by using an approximation argument. The real analogue was shown by Schoenberg in 1942, and the authors have given a new proof of Schoenberg's result by the method just described [Trans. Am. Math. Soc. 243, 89-95 (1978; Zbl 0394.15011)]. For similar results of C. S. Herz, D. Rider, C. C. Graham and W. Moran, replacing positive semidefinite matrices by positive definite functions on infinite abelian groups, see the book by \textit{C. C. Graham} and \textit{O. C. McGehee} [Essays in commutative harmonic analysis (1979; Zbl 0439.43001), Theorem 9.6.1].
scientific article; zbMATH DE number 3912978